# Quasisymmetric Embeddability of Weak Tangents

Wenbo Li  Department of Mathematics, Graduate Center, City University of New
York, NY, 10016,USA<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the quasisymmetric embeddability of weak tangents of
metric spaces. We first show that quasisymmetric embeddability is hereditary,
i.e., if $X$ can be quasisymmetrically embedded into $Y$, then every weak
tangent of $X$ can be quasisymmetrically embedded into some weak tangent of
$Y$, given that $X$ is proper and doubling. However, the converse is not true
in general; we will illustrate this with several counterexamples. In special
situations, we are able to show that the embeddability of weak tangents
implies global or local embeddability of the ambient space. Finally, we apply
our results to expanding dynamics and establish several results on Gromov
hyperbolic groups and visual spheres of expanding Thurston maps.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Notations and preliminaries
    2. 2.2 Gromov-Hausdorff distance and weak tangents of metric spaces
  3. 3 Quasisymmetric embeddability is hereditary
  4. 4 Counterexamples
    1. 4.1 Slit Sierpiński carpets
    2. 4.2 Rickman’s rugs
  5. 5 Self-quasisymmetricity and quasi-self-symmetricity
  6. 6 Expanding dynamics on metric spaces
    1. 6.1 Expanding maps
    2. 6.2 Gromov hyperbolic spaces and groups
    3. 6.3 Visual spheres of expanding Thurston maps

##  1\. Introduction

The Gromov-Hausdorff distance gives a precise meaning to how close or far
apart two arbitrary (compact) metric spaces are. The definition that is widely
used nowadays can be traced back to Gromov in [Gr81a] and [Gr81b]. More
precisely, the Gromov-Hausdorff distance of two compact metric spaces is the
infimum Hausdorff distance of their isometric images in the same space.
Intuitively, it measures how far the two compact metric spaces are from being
isometric.

The weak tangents of a metric space are analogous to the tangent planes of a
surface. Let $X$ be a metric space and $p\in X$. “Blowing up” $X$ near $p$
generates a sequence of dilations, which provides a better and better
illustration of the local behavior near $p$. A weak tangent at $p$ is the
limit (if it exists) of such a sequence, where the limit is in the (pointed)
Gromov-Hausdorff sense. Gromov’s compactness theorem shows that a
subconvergent metric space exists, given some conditions on the ambient space.
See Proposition 2.4 and 2.7. We denote by $WT_{p}(X)$ the collection of all
weak tangents at $p$ in $X$. See Section 2 for more details.

A natural question raised by weak tangents is: Is there any analytical or
geometrical relation between the ambient space and its weak tangents? There
are some answers to this question. One remarkable achievement in answering
this question is Cheeger-Colding theory. Cheeger-Colding theory investigates
the analytical and geometrical properties of the weak tangents of complete
connected manifolds with lower bounded Ricci curvatures. See [CC97],[CC00a]
and [CC00b] for more details. Cheeger-Colding theory is widely used in many
important works, including the proof of Thurston’s geometrization conjecture
and the proof of the existence of Kähler-Einstein metrics on Fano manifolds.
Weak tangents also play a role in studying Poincaré inequalities on metric
spaces, see [Ch99] and [CK15] for examples. In [BKM99] and [BM13], metric
fractals are studied together with weak tangents.

A homeomorphism $f:X\to Y$ is _$\eta$ -quasisymmetric_, where
$\eta:[0,\infty)\to[0,\infty)$ is a homeomorphism, if

$\frac{d_{Y}(f(x),f(y))}{d_{Y}(f(x),f(z))}\leq\eta\left(\frac{d_{X}(x,y)}{d_{X}(x,z)}\right)$

for all $x,y,z\in X$ with $x\neq z$. A quasisymmetry is a generalization of a
bi-Lipschitz map, and it preserves the approximate shape and the relative
size. See Section 2 for more details.

In this paper, we study the quasisymmetric embeddability of weak tangents of
metric spaces. We are interested in how the quasisymmetric embeddability of
the ambient space relates to that of its weak tangents.

######  Theorem 1.1.

Let $X,Y$ be proper, doubling metric spaces and $f:(X,p,d_{X})\to(Y,q,d_{Y})$
be an $\eta$-quasisymmetric map. For any weak tangent $T_{p}X\in WT_{p}(X)$,
there exists a weak tangent $T_{q}Y\in WT_{q}(Y)$ such that $T_{p}X$ is
$\eta$-quasisymmetric equivalent to $T_{q}Y$.

Theorem 1.1 shows that quasisymmetric embeddability is hereditary. Roughly
speaking, any quasisymmetric embedding between two metric spaces induces
quasisymmetric embeddings between their weak tangents. However, the converse
implication is not true in general. See Section 4 for several counterexamples.
In special situations, we are able to prove the following two converse
implications:

######  Theorem 1.2.

Let $X$ be a proper, doubling metric space and $p$ be a point in $X$. Assume
that there exists a sequence of points $\\{p_{n}\\}$ such that $X$ is
$\eta$-self-quasisymmetric at $p_{n}$ for every $n$ and $p_{n}\to p$ in $X$.
Then there exists a weak tangent $T_{p}X$ in $WT_{p}(X)$ such that $X$ is
$\eta^{\prime}$-quasisymmetrically embedded into $T_{p}X$, where
$\eta^{\prime}(t)=1/\eta^{-1}\\!\\!\left(\frac{1}{t}\right)$.

Theorem 1.2 induces the following result: If every weak tangent at $p$ is
uniformly quasisymmetrically embedded into a metric space $Y$, then $X$ is
also quasisymmetrically embedded into $Y$.

Roughly speaking, Theorem 1.2 shows that metric spaces that are “ self-
similar” in a quasisymmetric sense satisfy a global converse implication. We
can further strengthen Theorem 1.2 (i.e., every weak tangent at $p$ admits a
quasisymmetric image of $X$) by adding more conditions. See Corollary 5.1 and
5.2.

The following local embedding theorem gives another type of converse
implication:

######  Theorem 1.3.

Let $X$ be a compact, proper, doubling, $\eta$-quasi-self-symmetric space, $p$
be a point in $X$ and $T_{p}X$ be a weak tangent in $WT_{p}(X)$. There exists
a ball $B(q,r)$ in $X$ and an $\eta^{\prime}$-quasisymmetric embedding
$f:B(q,r)\to T_{p}X$ such that $f(q)=p_{\infty}$ and
$\eta^{\prime}(t)=1/\eta^{-1}\\!\\!\left(\frac{1}{t}\right)$. If $X$ is
uniformly perfect, $r$ depends only on $X$ and $\eta$.

Theorem 1.3 induces the following result: If every weak tangent at $p$ is
uniformly quasisymmetrically embedded into a metric space $Y$, then $B(q,r)$
is also quasisymmetrically embedded into $Y$.

Roughly speaking, Theorem 1.3 shows that metric spaces that are “quasi-self-
similar” in a quasisymmetric sense satisfy a local converse implication. These
terms that appeared in the above theorems are defined in Section 5.

Theorem 1.3 induces several applications.

A Kleinian group $\Gamma$ is a discrete subgroup of isometries of a hyperbolic
space and the limit set of $\Gamma$ is the set of accumulation points of the
orbit $\Gamma p$ of any element $p$ in the hyperbolic space. A Schottky set is
a compact subset of $\mathbb{S}^{2}$ whose complement is a union of at least
three round open disks whose closures are disjoint. See Subsection 6.2 for
more details.

The first application is a rigidity theorem for Kleinian groups whose limit
sets are Schottky sets.

######  Theorem 1.4.

Suppose that $\Gamma$ and $\tilde{\Gamma}$ are Kleinian groups whose limit
sets $S$ and $\tilde{S}$ are Schottky sets, respectively. We assume that
$\Gamma$ acts on $S$ and $\tilde{\Gamma}$ acts on $\tilde{S}$ as uniform
convergence groups. Let $p$, $\tilde{p}$ be points in $\Gamma$ and
$\tilde{\Gamma}$, respectively. If there exist a weak tangent at $p$ and a
weak tangent at $\tilde{p}$ such that they are quasisymmetrically equivalent,
then there exists a Möbius transformation mapping $S$ to $\tilde{S}$.

In simple words, Theorem 1.4 shows that a quasisymmetric equivalence between
any two weak tangents of the above spaces implies an equivalence of Möbius
transformations between the ambient spaces. Theorem 1.4 generalizes Theorem
$1.1$ in [Me14] to the situation of weak tangents.

An expanding Thurston map $f:S^{2}\to S^{2}$ is a postcritically-finite
branched covering map on a topological sphere with $\deg(f)\geq 2$ where $f$
locally expands $S^{2}$. A visual metric $\rho$ is a specific metric on
$S^{2}$ generated by $f$ such that $f$ locally expands $(S^{2},\rho)$ in a
uniform way. We call $(S^{2},\rho)$ is a visual sphere of $f$. See Subsection
6.3 for more details.

The other application is about the visual spheres of expanding Thurston maps:

######  Theorem 1.5.

Let $(S^{2},\rho)$ be a visual sphere of an expanding Thurston map $f:S^{2}\to
S^{2}$ that does not have periodic critical points. The following statements
are equivalent:

  1. (1)

$(S^{2},\rho)$ is a quasi-sphere

  2. (2)

Every weak tangent of $(S^{2},\rho)$ is quasisymmetric to $\mathbb{R}^{2}$.

  3. (3)

There exists a weak tangent of $(S^{2},\rho)$ that is quasisymmetric to
$\mathbb{R}^{2}$.

  4. (4)

There exists an open subset $U\subset S^{2}$ such that $U$ is
quasisymmetrically embedded into $\mathbb{R}^{2}$.

Theorem 1.5 illustrates a complete characterization of when visual spheres of
expanding Thurston maps are quasi-spheres. It has been proved as the main
theorem in [Wu19]. Wu proved this result with ideas from dynamics. Here we
give an alternate proof with the ideas generated in Section 5 of this paper.

The paper is organized as follows. In Section 2 we give basic definitions and
properties of Gromov-Hausdorff distance and weak tangents. In Section 3 we
prove Theorem 1.1 and show some results inspired by this theorem. However, the
converse is not true in general and we illustrate several counterexamples in
Section 4. Section 5 is devoted to proving Theorem 1.2 and Theorem 1.3 and
thus showing that in specific situations the embeddability of weak tangents
implies the embeddability(or local embeddability) of the ambient space. In the
last section, Section 6, we investigate metric spaces with expanding dynamics
and apply our results to them. We prove Theorem 1.4 and Theorem 1.5 in this
section.

##  2\. Preliminaries

###  2.1. Notations and preliminaries

Let $(X,d_{X})$ be a metric space. We denote by

$B(x,r)=\\{x^{\prime}\in
X:d_{X}(x,x^{\prime})<r\\}\qquad\textrm{and}\qquad\overline{B}(x,r)=\\{x^{\prime}\in
X:d_{X}(x,x^{\prime})\leq r\\}$

the open and closed ball centered at $a$ with radius $r$, respectively.
Furthermore, we denote by

$\partial B(x,r)=\overline{B}(x,r)\setminus B(x,r)=\\{x^{\prime}\in
X:d_{X}(x,x^{\prime})=r\\}$

the metric boundary of $B(a,r)$.

If $X^{\prime}$ is a subset of $X$, we denote by

$N_{r}(X^{\prime})=\\{x\in X:\exists\ x^{\prime}\in X^{\prime}\ \textrm{such
that}\ d_{X}(x,x^{\prime})<r\\}$

the $r$-neighborhood of $X^{\prime}$ for any $r>0$.

We denote by $\mathbb{S}^{n},\mathbb{D}^{n},\mathbb{H}^{n}$ the
$n$-dimensional unit sphere, unit ball and hyperbolic space, respectively.
Specifically, We denote by $|\ \cdot\ |$ the standard metric in Euclidean
spaces.

A homeomorphism $f:X\to Y$ is _(metrically) $H$-quasiconformal_ for some
$1\leq H<\infty$ if

$\displaystyle\limsup_{r\to 0}\frac{\sup\\{d_{Y}(f(x),f(y)):d_{X}(x,y)\leq
r\\}}{\inf\ \\{d_{Y}(f(x),f(y)):d_{X}(x,y)\geq r\\}}\leq H$

for $\forall x\in X$. A map is quasiconformal if it is $H$-quasiconformal for
some $H$.

A homeomorphism $f:X\to Y$ is _$\eta$ -quasisymmetric_, where
$\eta:[0,\infty)\to[0,\infty)$ is a homeomorphism, if

$\frac{d_{Y}(f(x),f(y))}{d_{Y}(f(x),f(z))}\leq\eta\left(\frac{d_{X}(x,y)}{d_{X}(x,z)}\right)$

for all $x,y,z\in X$ with $x\neq z$. The map $f$ is called _quasisymmetric_ if
it is $\eta$-quasisymmetric for some _distortion function_ $\eta$.

Here are some useful properties of quasisymmetric maps, which will be used
repeatedly in the paper.

######  Proposition 2.1.

Suppose $f:X\to Y$ and $g:Y\to Z$ are $\eta$ and $\theta$-quasisymmetric
mappings, respectively.

  1. (1)

The composition $g\circ f:X\to Z$ is an $\theta\circ\eta$-quasisymmetric map.

  2. (2)

The inverse $f^{-1}:Y\to X$ is an $\eta^{\prime}$-quasisymmetric map, where
$\eta^{\prime}(t)=1/\eta^{-1}\\!\\!\left(\frac{1}{t}\right)$.

  3. (3)

If $A$ and $B$ are bounded subsets of $X$ and $A\subset B$, then

$\frac{1}{2\eta\left(\frac{\mathrm{diam}(B)}{\mathrm{diam}(A)}\right)}\leq\frac{\mathrm{diam}(f(A))}{\mathrm{diam}(f(B))}\leq\eta\left(\frac{2\mathrm{diam}(A)}{\mathrm{diam}(B)}\right).$

A metric space is called _proper_ if the closure of every ball is compact.

A metric space is called _doubling_ if there exists a universal constant
$C\geq 1$ such that every subset of diameter $d$ can be covered by at most $C$
subsets of diameter at most $d/2$. Notice that a space is doubling implies
that it is separable, due to the Assouad’s embedding Theorem. See Theorem
$12.1$ and $12.2$ in [He01].

A metric space $X$ is called _uniformly perfect_ if there exists a universal
constant $C\geq 1$ such that for each $x\in X$ and for each $r>0$,
$B(x,r)\setminus\overline{B}(x,r/C)$ is nonempty whenever $X\setminus B(x,r)$
is nonempty. Uniformly perfectness forbids isolated islands in a uniformly
manner. In other words, a metric space is uniformly perfect if for every empty
metric annulus in the space, the ratio between the outer and inner radii is
bounded from above.

A metric space is equipped with the _intrinsic metric_ if the distance of any
two points equals the infimum length of all the paths joining them. A metric
space whose metric is intrinsic is called a _length space_. A _geodesic_ in a
length space is a curve which is locally a distance minimizer(i.e., a shortest
path).

###  2.2. Gromov-Hausdorff distance and weak tangents of metric spaces

Recall that the _Hausdorff distance_ between two subsets $X$ and $Y$ in the
same ambient space, denoted by $d_{H}(X,Y)$, is defined by

(1) $d_{H}(X,Y):=\inf\\{r>0:X\subset N_{r}(Y)\ \textrm{and}\ Y\subset
N_{r}(X)\\}.$

The _Gromov-Hausdorff distance_ $d_{GH}(X,Y)$ of two metric space $X$ and $Y$
(do not need to be in the same space), which largely extend the definition of
Hausdorff distance, is defined by

(2) $d_{GH}(X,Y):=\inf_{f,g}\\{d_{H}(f(X),g(Y))\\}.$

where $f:X\to Z$ and $g:Y\to Z$ are isometric embeddings into some metric
space $Z$.

The following theorem shows that Gromov-Hausdorff distance defines a metric on
a certain space.

######  Theorem 2.2.

Let $\mathfrak{X}$ be the family of all compact metric spaces, then Gromov-
Hausdorff distance defines a finite metric on its isometry classes. More
precisely, $d_{GH}$ is finite, nonnegative, symmetric, satisfy the triangle
inequality and $d_{GH}(X,Y)=0$ iff $X$ is isometric to $Y$.

A sequence of compact metric spaces $\\{X_{n}\\}_{n=1}^{\infty}$ _Gromov-
Hausdorff converges_ to a compact metric space $X$ if $d_{GH}(X_{n},X)\to 0$
as $n\to\infty$. We denote it by $X_{n}\xrightarrow{GH}X$. Notice that the
limit space is unique up to isometries. It is easy to see that $X_{n}$
Hausdorff converges to $X$ implies $X_{n}\xrightarrow{GH}X$.

A map $f:X\to Y$ is called an _$\epsilon$ -isometry_ if

$\mathrm{dist}(f):=\sup\\{|d_{X}(x_{1},x_{2})-d_{Y}(f(x_{1}),f(x_{2})|:x_{1},x_{2}\in
X\ \\}\leq\epsilon$

and

$d_{H}(f(X),Y)\leq\epsilon.$

The following propositions play an important role when dealing with Gromov-
Hausdorff convergence. See Chapter $7$ of [BBI01] for more details.

######  Proposition 2.3.

Let $X$ and $Y$ be two metric spaces and $\epsilon>0$.

  1. (1)

If $d_{GH}(X,Y)<\epsilon$, then there exists a $2\epsilon$-isometry from $X$
to $Y$.

  2. (2)

If there exists an $\epsilon$-isometry from $X$ to $Y$, then
$d_{GH}(X,Y)<2\epsilon$.

######  Proposition 2.4 (Precompactness).

Let $\mathfrak{X}$ be a collection of compact, uniformly bounded and doubling
metric spaces. If the doubling constant of every element in $\mathfrak{X}$ is
uniformly bounded, then $\mathfrak{X}$ is precompact in the Gromov-Hausdorff
topology, i.e., any sequence of elements of $\mathfrak{X}$ contains a
convergent subsequence in the Gromov-Hausdorff sense.

A _pointed metric space_ is a triple $(X,p,d_{X})$ where $(X,d_{X})$ is a
metric space with a base point $p\in(X,d_{X})$. The _pointed Gromov-Hausdorff
convergence_ is an analog of Gromov-Hausdorff convergence appropriate for non-
compact spaces. A sequence of metric spaces $\\{(X_{n},p_{n},d_{n})\\}$
_pointed Gromov-Hausdorff converges_ to a complete metric space $(X,p,d_{X})$
if for every $r>0$ and $\epsilon>0$ there exists a natural number $n_{0}$ such
that for every $n>n_{0}$ there exists a (not necessarily continuous) map
$f:B(p_{n},r)\to X$ such that the following hold:

  1. (1)

$f(p_{n})=p$;

  2. (2)

$\mathrm{dist}(f)<\epsilon$;

  3. (3)

$B(p,r-\epsilon)\subset N_{\epsilon}(f(B(p_{n},r)))$.

We also denote it by $(X_{n},p_{n},d_{n})\xrightarrow{GH}(X,p,d_{X})$. Readers
can verify that there is no difference between open and closed balls in the
definition. Intuitively, the ball $B(p_{n},r)$ in $X_{n}$ lies within the
Gromov-Hausdorff distance of order $\epsilon$ from a subset of $X$ between the
ball of radii $r-\epsilon$ and $r+2\epsilon$ centered at $p$.

Notice that the map $f:B(p_{n},r)\to B(p,r+\epsilon)$ is a
$3\epsilon$-isometry and $f(p)=p_{n}$. We call a map which is an
$\epsilon$-isometry and keeps the base point a _pointed $\epsilon$-isometry_.
Similarly, it can be directly verified that there exists a pointed
$6\epsilon$-isometry $f^{\prime}:B(p,r)\to B(p_{n},r+2\epsilon)$ such that
$f^{\prime}(p)=p_{n}$.

Figure 1. An example of pointed Gromov Hausdorff convergence

The definition of pointed Gromov-Hausdorff convergence varies in different
resources. We choose the one which balances intuition and generality. If $X$
is a length space, then our definition of Gromov-Hausdorff convergence implies
the following one: $\overline{B}(p_{n},r)\xrightarrow{GH}\overline{B}(p,r)$
for every $r>0$. Notice that we have abused notation slightly here. Although
$\overline{B}(p_{n},r)$ may not be compact; however, we can still show that
$d_{GH}(\overline{B}(p_{n},r),\overline{B}(p,r))\to 0$. Readers can find more
information in Chapter $7$ and $8$ of [BBI01].

The following proposition shows that Gromov-Hausdorff convergence is
equivalent to pointed Gromov-Hausdorff convergence for compact metric spaces.

######  Proposition 2.5.

Let $(X,d)$ and $\\{(X_{n},d_{n})\\}$ be compact metric spaces, then

  1. (1)

If $(X_{n},p_{n},d_{n})\xrightarrow{GH}(X,p,d)$, then
$X_{n}\xrightarrow{GH}X$.

  2. (2)

If $X_{n}\xrightarrow{GH}X$ and $p\in X$, then there exists a sequence of
$p_{n}\in X_{n}$ such that $(X_{n},p_{n},d_{n})\xrightarrow{GH}(X,p,d)$.

We denote by $f:(X,p,d_{X})\to(Y,q,d_{Y})$ a map between two pointed metric
space which keeps the base points. Namely, $f$ is a map between $(X,d_{X})$
and $(Y,d_{Y})$ and $f(p)=q$.

Let ${(X_{n},p_{n},d_{n})}$ and ${(Y_{n},q_{n},l_{n})}$ be sequences of
pointed metric spaces that pointed Gromov-Hausdorff converge to $(X,p,d)$ and
$(Y,q,l)$, respectively. Let ${f_{n}}$ be a sequence of mappings given by
$f_{n}:(X_{n},p_{n},d_{n})\to(Y_{n},q_{n},l_{n})$ for each $n\in\mathbb{N}$
and let $f:(X,p,d)\to(Y,q,l)$. We say _$f_{n}$ converges to $f$_ if there
exist pointed $\epsilon_{n}$-isometries

$\varphi^{n}:X\to X_{n}\qquad\textrm{and}\qquad\psi_{n}:Y_{n}\to Y$

such that the following holds:

$\lim_{n\to\infty}\psi_{n}\circ f_{n}\circ\varphi^{n}(x)=f(x)$

for any $x\in X$ and $\epsilon_{n}\to 0$ as $n\to\infty$. Naturally, we have
$f(p)=q$.

Let $(X,d_{X})$ be a metric space and $p$ be a point in $X$. We call a pointed
metric space a _weak tangent_ of $X$ at $p$ if it is the pointed Gromov-
Hausdorff limit of a sequence of spaces
$\\{(X,p_{n},\frac{d_{X}}{\lambda_{n}})\\}$ where $\lambda_{n}\to 0$ and
$p_{n}\to p$ in $X$. We denote by $(T_{p}X,p_{\infty},g_{p})$ the above weak
tangent. A specific case of the weak tangents is when $p_{n}=p$ for infinitely
many $n$. Notice that, if the weak tangent exists, it is equivalent to that
$p_{n}=p$ for all $n$. We call such a weak tangent a _proper weak tangent_ at
$p$.

One should be aware that there may exist more than one weak tangent at the
same point.

######  Proposition 2.6.

Let $\\{(X_{n},p_{n},d_{n})\\}$ be a sequence of pointed metric spaces and
$(X_{n},p_{n},d_{n})\xrightarrow{GH}(X,p,d)$, then
$(X_{n},p_{n},\frac{d_{n}}{\lambda})\xrightarrow{GH}(X,p,\frac{d}{\lambda})$
for any $\lambda>0$.

We denote by $WT_{p}(X)$ and $WT(X)$ the collections of all weak tangents at
point $p$ and all weak tangents of $X$, respectively. Similarly, we denote by
$PWT_{p}(X)$ and $PWT(X)$ the collections of all proper weak tangents at point
$p$ and all proper weak tangents of $X$, respectively.

A natural question is whether there exists at least one weak tangent at every
point. The answer is given in the following proposition.

######  Proposition 2.7.

Let $\mathfrak{X}$ be a collection of pointed metric spaces. Suppose that for
every $r>0$ there exists a constant $C$ depending on $r$ such that for every
$(X,p)\in\mathfrak{X}$, the ball $B(p,r)$ is $C$-doubling, then any sequence
of elements of $\mathfrak{X}$ contains a convergent subsequence in the point
Gromov-Hausdorff sense.

Proposition 2.7 is analogous to Proposition 2.4. Roughly speaking, it requires
the property of uniformly local doubling to induce precompactness.

##  3\. Quasisymmetric embeddability is hereditary

In this section we first prove Theorem 1.1, then establish the following
result: If $X$ is quasisymmetrically embedded into a normed vector space, then
every element in $WT(X)$ is quasisymmetrically embedded into the same space,
given that $X$ is proper and doubling.

The proof of Theorem 1.1 is mainly finished by next Lemma. See Lemma 2.4.7 in
[KL04].

######  Lemma 3.1 (Keith, Laakso).

Let ${(X_{n},p_{n},d_{n})}$ and ${(Y_{n},q_{n},l_{n})}$ be sequences of proper
pointed metric spaces that pointed Gromov-Hausdorff converge to $(X,p,d)$ and
$(Y,q,l)$, respectively. Let $f_{n}:(X_{n},p_{n})\to(Y_{n},q_{n})$ be an
$\eta$-quasisymmetric map for each $n\in\mathbb{N}$, where $\eta$ is fixed. If
there exist a $C\geq 1$ and a sequence $\\{x_{n}\\}$, where each $x_{n}\in
X_{n}$, such that

(3) $\frac{1}{C}\leq d_{n}(p_{n},x_{n})\leq
C\qquad\textrm{and}\qquad\frac{1}{C}\leq l_{n}(q_{n},f(x_{n}))\leq C$

for every $n\in\mathbb{N}$. Then, after passing to a subsequence, we have
$\\{f_{n}\\}$ converges to some $\eta$-quasisymmetric map $f$ between $X$ and
$Y$ and $f(p)=q$.

The proof is based on the equicontinuity of quasisymmetric maps and the
Arzelá-Ascoli theorem. For the sake of completeness, we will illustrate a
concise proof of the above lemma. Notice that our definitions of pointed
Gromov-Hausdorff convergence and weak tangent are different as that in [KL04];
however, the idea is still the same. We first introduce two definitions.

A family of mappings $f_{n}:(X_{n},p_{n},d_{n})\to(Y_{n},q_{n},l_{n})$ are
_equicontinuous on bounded subsets_ if for any $R>0$, and for any
$\epsilon>0$, there exists a $\delta>0$ such that

$l_{n}(f_{n}(x),f_{n}(x^{\prime}))<\epsilon\ \textrm{whenever}\
x,x^{\prime}\in B(p_{n},R)\ \textrm{and}\ d_{n}(x,x^{\prime})<\delta.$

A family of mappings $f_{n}:(X_{n},p_{n},d_{n})\to(Y_{n},q_{n},l_{n})$ are
_uniformly bounded on bounded sets_ if for any $R>0$,

$\sup_{n}\sup_{x,x^{\prime}\in
B(p_{n},R)}l_{n}(f_{n}(x),f_{n}(x^{\prime}))<\infty.$

###### Proof of Lemma 3.1.

$\\{f_{n}\\}$ are equicontinuous on bounded subsets due to inequalities (3).
The proof is analogous to Proposition $10.26$ in [He01]. Similarly,
inequalities (3) implies that $\\{f_{n}\\}$ are uniformly bounded on bounded
sets.

Assume that $X$ and $Y$ are bounded; thus it can be directly verified that
$X_{n}$ and $Y_{n}$ should be uniformly bounded. Let’s see the following
diagram:

|
---|---
$\textstyle{X_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{n}}$$\textstyle{Y_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{n}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\varphi^{n}}$$\textstyle{Y.}$

In this diagram, $\varphi^{n}$ and $\psi_{n}$ are pointed
$\frac{1}{n}$-isometries from $X$ to $X_{n}$ and $Y_{n}$ to $Y$, respectively.
We will construct a function $f$ such that the above diagram commutes when
$n\to\infty$.

Since $X$ is doubling, it is separable; thus we can select a dense countable
subset $E$ of $X$. For any $x_{1}\in E$, the sequence $\psi_{n}\circ
f_{n}\circ\varphi^{n}(x_{1})$ is bounded since $f_{n}$ is uniformly bounded on
bounded sets. $Y$ is a proper space; thus there exists a subsequence $n_{i}$
such that $\psi_{n_{i}}\circ f_{n_{i}}\circ\varphi^{n_{i}}(x_{1})$ converges
to a point $y_{1}$ in $Y$. We define $f(x_{1}):=y_{1}$. By Cantor’s diagonal
argument, there exists a subsequence $\\{n_{j}\\}$ such that we are able to
define a function $f$ on $E$ where

$f(x):=\lim_{j\to\infty}\psi_{n_{j}}\circ f_{n_{j}}\circ\varphi^{n_{j}}(x)$

for any $x\in E$. Combining with the equicontinuity on bounded subsets, we can
follow a proof analogous to the Arzelá-Ascoli theorem to define a function
$f:(X,p,d)\to(Y,q,l)$ as the limit of $\\{f_{n_{j}}\\}$. It requires some
basic tricks of pointed $\epsilon$-isometries(See Proposition 2.3) to show
that $f$ is a quasisymmetry. We leave it to the readers.

If $X$ and $Y$ are unbounded, we select exhaustions $\\{U_{n}\\}$ and
$\\{V_{n}\\}$ of $X$ and $Y$(namely, a sequence of bounded subsets of the
ambient space with the union equals to the whole space), respectively, such
that each $U_{n}$ is $\eta$-quasisymmetric equivalent to $V_{n}$. Applying
Cantor’s diagonal argument again, we finish the proof by taking a sub-limit. ∎

###### Proof of Theorem 1.1.

Let $\\{(X,p_{n},\frac{d_{X}}{\lambda_{n}})\\}$ converges to $T_{p}X$ and
$f(p_{n})=q_{n}$. We split the proof into two situations: either $X$ is
uniformly perfect or not.

If $X$ is uniformly perfect, then there exists a constant $C\geq 1$ and a
sequence of $x_{n}\in X$ such that

(4) $\frac{\lambda_{n}}{C}\leq d_{X}(p_{n},x_{n})\leq\lambda_{n}.$

Let $\omega_{n}=d_{Y}(q_{n},f(x_{n}))$. Thus we define
$f_{n}:(X,p_{n},\frac{d_{X}}{\lambda_{n}})\to(Y,q_{n},\frac{d_{Y}}{\omega_{n}})$
by $f_{n}(x):=f(x)$. Notice that $\omega_{n}\neq 0$, $\omega_{n}\to 0$ as
$n\to\infty$ and $f_{n}$ are still $\eta$-quasisymmetric.

Apply Lemma 3.1 on $\\{f_{n}\\}$ with the following inequalities

(5) $\frac{1}{C}\leq\frac{d_{X}}{\lambda_{n}}(p_{n},x_{n})\leq
1\qquad\textrm{and}\qquad\frac{d_{Y}}{\omega_{n}}(q_{n},f_{n}(x_{n}))=1$

finishes the proof.

If $X$ is not uniformly perfect, we split the proof into two cases.

Case $1$. Notice that $\\{p_{n}\\}$ and $\\{\lambda_{n}\\}$ are fixed. If
inequality (4) exists for some sequence $\\{x_{n}\\}$ and some constant $C\geq
1$, then it finishes the proof.

Case $2$. If not, every sequence of $x_{n}\in X$ should satisfies that
$\frac{d_{X}}{\lambda_{n}}(p_{n},x_{n})\to 0$ when
$\frac{d_{X}}{\lambda_{n}}(p_{n},x_{n})$ is uniformly bounded from above and
$\frac{d_{X}}{\lambda_{n}}(p_{n},x_{n})\to\infty$ when
$\frac{d_{X}}{\lambda_{n}}(p_{n},x_{n})$ is uniformly bounded from below.

For any sequence $\\{a_{n}\\}$ where each $a_{n}\in B(p_{n},\lambda_{n})$, we
have $\frac{d_{X}}{\lambda_{n}}(p_{n},a_{n})<1$ for every $n$. Thus
$\frac{d_{X}}{\lambda_{n}}(p_{n},a_{n})\to 0$.

For any sequence $\\{b_{n}\\}$ where each $b_{n}\in
X\setminus\overline{B}(p_{n},\lambda_{n})$, we have
$\frac{d_{X}}{\lambda_{n}}(p_{n},b_{n})>1$ for every $n$. Thus
$\frac{d_{X}}{\lambda_{n}}(p_{n},b_{n})\to\infty$.

$\partial B(P_{n},\lambda_{n})$ should be empty when $n$ is sufficiently
large, otherwise inequality (4) exists.

Thus $(X,p_{n},\frac{d_{X}}{\lambda_{n}})$ is inside the complement of a
metric torus $B(p_{n},R_{n})\setminus\overline{B}(p_{n},r_{n})$ where
$\lim\limits_{n\to\infty}R_{n}=\infty$ and $\lim\limits_{n\to\infty}r_{n}=0$.
It is trivial to show that
$(X,p_{n},\frac{d_{X}}{\lambda_{n}})\xrightarrow{GH}\\{p\\}$. ∎

Lemma 3.1 and Theorem 1.1 are still valid in some other situations. Namely,
they are also valid for maps that are isometric, i.e.,
$l_{n}(f_{n}(x),f_{n}(y))=d_{n}(x,y)$, $r$-similar, i.e.,
$l_{n}(f_{n}(x),f_{n}(y))=r\cdot d_{n}(x,y)$, and $L$-bilipschitz, i.e.,
$1/L\cdot d_{n}(x,y)\leq l_{n}(f_{n}(x),f_{n}(y))\leq L\cdot d_{n}(x,y)$. The
proof are verbatim the same.

######  Lemma 3.2.

Let ${(X_{n},p_{n},d_{n})}$ and ${(Y_{n},q_{n},l_{n})}$ be sequences of proper
pointed metric spaces that pointed Gromov-Hausdorff converge to $(X,p,d)$ and
$(Y,q,l)$, respectively. Let $f_{n}:(X_{n},p_{n})\to(Y_{n},q_{n})$ be an
isometric/$r$-similar/$L$-bilipschitz/$\eta$-quasisymmetric map for each
$n\in\mathbb{N}$. If there exist a $C\geq 1$ and a sequence $\\{x_{n}\\}$,
where each $x_{n}\in X_{n}$, such that

(6) $\frac{1}{C}\leq d_{n}(p_{n},x_{n})\leq
C\qquad\textrm{and}\qquad\frac{1}{C}\leq l_{n}(q_{n},f(x_{n}))\leq C$

for every $n\in\mathbb{N}$. Then, after passing to a subsequence, we have
$\\{f_{n}\\}$ converges to some
isometric/$r$-similar/$L$-bilipschitz/$\eta$-quasisymmetric map $f$ between
$X$ and $Y$ and $f(p)=q$.

######  Theorem 3.3.

Let $X,Y$ be proper, doubling metric spaces and $f:(X,p,d_{X})\to(Y,q,d_{Y})$
be an an isometric/$r$-similar/$L$-bilipschitz/$\eta$-quasisymmetric map. For
any $T_{p}X\in WT_{p}(X)$, there exists a $T_{q}Y\in WT_{q}(Y)$ such that
$T_{p}X$ is isometric/$r$-similar/$L$-bilipschitz/$\eta$-quasisymmetric
equivalent to $T_{q}Y$.

We are interested in a metric space which admits all weak tangents of its
subsets, i.e., a metric space $X$ with a property that every weak tangent of
every $Y\subset X$ can be uniformly quasisymmetrically embedded into $X$. We
call such a space _quasisymmetric-tangent-self-embeddable_. The following
lemma proves that a finite dimensional normed vector space admits this
property.

######  Lemma 3.4.

Let $Y$ be a subset of a finite dimensional normed vector space $(X,d)$, then
every weak tangent of $Y$ is (isometrically) a subset of $X$.

###### Proof.

Since $X$ is finite dimensional, it is doubling and proper. In fact, doubling
and proper are equivalent to finite dimension here. Let $T_{p}Y$ be a weak
tangent of $Y$ and
$(Y,p_{n},\frac{d}{\lambda_{n}})\xrightarrow{GH}(T_{p}Y,p_{\infty},g_{p})$.
Let

$\lambda_{n}Y=\\{x\in X:\textrm{there exists a}\ y\in Y\ \textrm{such that}\
x-p_{n}=\lambda_{n}(y-p_{n})\\}$

be a linear dilation of $Y$ at point $p_{n}$. Thus $(\lambda_{n}Y,p_{n},d)$ is
isometric to $(Y,p,\frac{d}{\lambda_{n}})$ for any $\lambda_{n}$.

We claim that there exists a subsequence $\\{\lambda_{n_{i}}\\}$ and a
$Y_{p}\subset X$ such that for every $r>0$,
$\overline{B}(p_{n_{i}},r)\cap\lambda_{n_{i}}Y$ Hausdorff converges to
$\overline{B}(p,r)\cap Y_{p}$. The proof is analogous to Proposition 2.7. This
is achieved by Theorem $7.3.8$ in [BBI01] and Cantor’s diagonal argument. Thus
$(\lambda_{n_{i}}Y,p_{n_{i}},d)\xrightarrow{GH}(Y_{p},p,d)$. Since
$(Y,p,\frac{d}{\lambda_{n_{i}}})\xrightarrow{GH}(T_{p}Y,p_{\infty},g_{p})$, we
finish the proof by Lemma 3.2. ∎

######  Corollary 3.5.

Let $X$ be a proper, doubling metric space and $V$ be a normed vector space.
If $X$ can be $\eta$-quasisymmetrically embedded into $V$, then any weak
tangent of $X$ can be $\eta$-quasisymmetrically embedded into $V$.

Since $X$ is proper and doubling, its quasisymmetric image should be contained
in a finite dimensional subspace of $V$. The rest of the proof follows Theorem
1.1 and Lemma 3.4.

Corollary 3.5 is still valid if, for a normed vector space, we substitute any
quasisymmetric-tangent-self-embeddable space(with a little revision of the
proof). However, we know very few of these spaces. Finite dimensional normed
vector spaces are one, and it is plausible that any self-similar space with a
suitable “extension” to infinity will be another.

##  4\. Counterexamples

In this section, we will illustrate several counterexamples to show that the
converse of Theorem 1.1 is not true.

###  4.1. Slit Sierpiński carpets

Let $R=(a_{1},b_{1})\times(a_{2},b_{2})\subset\mathbb{R}^{2}$ be a rectangle
and let $s=\\{x\\}\times[c_{2},d_{2}]\subset R$ be a _slit_ in $R$, where
$l(s)=\mbox{diam}(s)=d_{2}-c_{2}$ is the _length_ of $s$.
$R\backslash\bigcup_{i=1}^{m}s_{i}$, where $s_{i}$’s are mutually disjoint
slits in $R$, is called a _slit domain_.

Given an infinite sequence $\mathbf{r}=\\{r_{i}\\}_{i=0}^{\infty}$ such that
$r_{i}\in(0,\sigma)$ for some $0<\sigma<1$, we define a slit domain

$\mathcal{S}_{n}(\mathbf{r}):=[0,1]^{2}\backslash\left(\bigcup_{i=0}^{n}\bigcup_{j=1}^{2^{i}}s_{ij}\right),$

where

  1. (1)

$s_{ij}\subset\Delta_{ij}$, where $\Delta_{ij}$ is a dyadic square of
generation $i$.

  2. (2)

The center of $s_{ij}$ coincides with the center of $\Delta_{ij}$.

  3. (3)

$l(s_{ij_{1}})=l(s_{ij_{2}})=r_{i}\cdot\frac{1}{2^{i}}$ for
$j_{1},j_{2}\in\\{1,\ldots,2^{i}\\}$.

Figure 2. Slit domains with respect to
$(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$ and
$(\frac{1}{10},\frac{2}{5},\frac{1}{8},\frac{1}{2})$

We equip $\mathcal{S}_{n}(\mathbf{r})$ with the intrinsic metric and denote by
$\mathscr{S}_{n}(\mathbf{r})$ the closure of $\mathcal{S}_{n}(\mathbf{r})$.
The Gromov-Hausdorff limit

$\mathscr{S}(\mathbf{r}):=\lim_{n\to\infty}\mathscr{S}_{n}(\mathbf{r})$

is called the _dyadic slit Sierpiński carpet with respect to $\mathbf{r}$_.
Notice that $\mathscr{S}(\mathbf{r})$ is a metric Sierpiński carpet. When we
mention a _slit_ in $\mathscr{S}(\mathbf{r})$, we mean the boundary component
of $\mathscr{S}(\mathbf{r})$ which is corresponding to a slit in the slit
domain.

The following theorem from [HL19] classifies the planar embeddability of
dyadic slit carpets.

######  Theorem 4.1.

Let $\mathscr{S}({\mathbf{r}})$ be a dyadic slit carpet with respect to
$\mathbf{r}$. There exists a quasisymmetric embedding of
$\mathscr{S}({\mathbf{r}})$ into $\mathbb{R}^{2}$ if and only if
$\mathbf{r}=\\{r_{i}\\}_{i=0}^{\infty}\in\ell^{2}$.

We call a dyadic slit carpet $\mathscr{S}(\mathbf{r})$ _harmonic_ if
$\\{r_{i}\\}\notin\ell^{2}$ and $r_{i}\to 0$ as $i\to\infty$. For example, a
dyadic slit carpet with respect to $\mathbf{r}=\\{\frac{1}{\sqrt{i}}\\}$ is
harmonic. Theorem 4.1 implies that there exists no quasisymmetric embedding
from a harmonic slit carpet to $\mathbb{R}^{2}$.

######  Proposition 4.2.

Any weak tangent of a harmonic $\mathscr{S}(\mathbf{r})$ is either the closed
one quarter plane $Q$, the closed half plane $H$, $\mathbb{R}^{2}$ or $T$,
where $T$ is the closure of $\mathbb{R}^{2}\setminus(0,\infty)$ which is
equipped with the intrinsic metric.

The proof of Proposition 4.2 is rather intuitive, so we just scratch the idea
here. A point $p$ in a metric carpet $S$ is _porous_ if there exists a
constant $C>1$ depending on $p$ such that for any $0<r<\mathrm{diam}(X)$,
there exists a boundary component $\Lambda$ of $S$ in $B(p,r)$ such that

$\frac{1}{C}<\frac{\mathrm{diam}(\Lambda)}{\mathrm{diam}(B(p,r))}<C.$

The proof of Proposition 4.2 is based on the idea that if
$\mathscr{S}(\mathbf{r})$ is harmonic, then no point in
$\mathscr{S}(\mathbf{r})$ is porous. In simple words, for any
$p\in\mathscr{S}(\mathbf{r})$ and any $0<r<\mathrm{diam}(X)$, the diameter of
any slit inside $B(p,r)$ will decrease to $0$ as “blowing up” the carpet near
$p$; thus “erase” all the slits inside $B(p,r)$.

Finally, it implies that any weak tangent at a point
$p\in\mathscr{S}(\mathbf{r})$ that doesn’t lie on any slit should be
$\mathbb{R}^{2}$. Readers can verify it with the tricks of
$\epsilon$-isometries(see Proposition 2.3). Similarly, any weak tangent at an
endpoint of a slit is $T$, and any weak tangent at a point on a slit but is
not the endpoint is $H$. There are still two cases left: any weak tangent at a
corner point of $\mathscr{S}(\mathbf{r})$ is $Q$ and any weak tangent at a
point on the outer boundary(largest boundary) of $\mathscr{S}(\mathbf{r})$ but
is not a corner point is $H$.

######  Corollary 4.3.

There exists a metric Sierpiński carpet such that it is not quasisymmetrically
embedded into $\mathbb{R}^{2}$ but every weak tangent is uniformly
quasisymmetrically embedded into $\mathbb{R}^{2}$.

###### Proof.

Let’s take a harmonic dyadic slit carpet and what is left to show is the
embeddability of $T$. We define a function $\Phi:H\to T$, which identifies the
points by the square map, and by showing it is a quasisymmetry we finish the
proof. More precisely, we define

$\displaystyle\Phi:H$ $\displaystyle\longrightarrow$ $\displaystyle T$
$\displaystyle re^{i\theta}$ $\displaystyle\mapsto$ $\displaystyle
r^{2}e^{2i\theta}$

for $r\geq 0$ and $\theta\in(0,\pi)$. However, we do require that
$\Phi(re^{i\pi})\neq\Phi(re^{i0})$ for $r\neq 0$. Since $T$ has two components
$R_{1}$ and $R_{2}$ of its boundary, see Figure 3, and both represent
$[0,+\infty)$, for any $r\geq 0$ we define $\Phi(re^{i0})\in R_{1}$ and
$\Phi(re^{i\pi})\in R_{2}$ in a continuous and symmetric way.

Figure 3. $\Phi:H\longrightarrow T$

It can be directly verified that $\Phi$ is a homeomorphism and, furthermore,
it is a conformal ($1$-quasiconformal) map. Since $H$ is a Loewner space, $T$
is linearly locally connected and they are both Ahlfors $2$-regular, $\Phi$ is
quasisymmetric by Corollary 4.8 in [HK98]. For the definition of Ahlfors
regularity, Loewner space and linearly locally connectivity, see Section $1$
and $3$ in [HK98] for a reference. ∎

Let $\mathbf{S}=\\{s_{i}\\}_{i=0}^{\infty}$ be the collection of all slits in
$\mathscr{S}(\mathbf{r})$. Let $\mathscr{P}_{i}$ be a metric space generated
by gluing three sides of two identical squares with sidelength $l(s_{i})$
identically. See Figure 4. $\mathscr{P}_{i}$ looks like a “square pillow” with
an “open mouth”. We define $\mathscr{R}(\mathbf{r})$ as

(7)
$\mathscr{R}(\mathbf{r}):=\mathscr{S}\bigsqcup_{\mathbf{G}}\\{\mathscr{P}_{i}\\}_{i=0}^{\infty},$

where $\mathbf{G}=\\{g_{i}\\}$ is a sequence of gluing functions. Each $g_{i}$
in $\mathbf{G}$ glues the slit $s_{i}$ with the boundary
$\partial\mathscr{P}_{i}$ identically. Notice that $\mathscr{R}(\mathbf{r})$
is topologically a unit square. We still equip the intrinsic metric on
$\mathscr{R}(\mathbf{r})$.

Figure 4. “Square pillow” $\mathscr{P}_{i}$, gluing function $g_{i}$ and slit
$s_{i}$.

It can be directly verified that $\mathscr{S}(\mathbf{r})$ is a (metrically)
subspace of $\mathscr{R}(\mathbf{r})$. Similarly, we call
$\mathscr{R}(\mathbf{r})$ _harmonic_ if the corresponding
$\mathscr{S}(\mathbf{r})$ is harmonic. Thus any harmonic
$\mathscr{R}(\mathbf{r})$ can not be quasisymmetrically embedded into
$\mathbb{R}^{2}$.

######  Proposition 4.4.

Any weak tangent of a harmonic $\mathscr{R}(\mathbf{r})$ is either the closed
one quarter plane $Q$, the closed half plane $H$, $\mathbb{R}^{2}$, $L$ or $D$
where

$L=T\bigsqcup_{l}H\qquad\textrm{and}\qquad D=Q\bigsqcup_{d}Q.$

$L$ is a length space where $l$ glues $\partial T$ with $\partial H$
identically. $D$ is a length space where $d$ glues $\partial Q$ with $\partial
Q$ identically.

The proof of Proposition 4.4 follows the same idea of the proof of Proposition
4.2. We leave it to the readers.

Recall that a _quasi-plane_ or a _quasi-sphere_ is a metric space which is
quasisymmetric equivalent to $\mathbb{R}^{2}$ or $\mathbb{S}^{2}$,
respectively. The following corollary provides counterexamples to the converse
implication.

######  Corollary 4.5.

Let $X$ be a topological plane or a topological sphere. There exist a metric
on $X$ such that $X$ is not a quasi-plane or a quasi-sphere, respectively, but
every weak tangent of $X$ is quasisymmetric equivalent to $\mathbb{R}^{2}$.

If replacing every unit square in $\mathbb{R}^{2}$ by $\mathscr{R}$ and equip
it with the intrinsic metric, we have the desired metric plane. Similarly, if
gluing two $\mathscr{R}$ isometrically by their outer boundaries and equip it
with the intrinsic metric, we have the desired metric sphere. Thus in order to
finish the proof of Corollary 4.5, it remains to show that $L$ and $D$ are
quasisymmetric equivalent to $\mathbb{R}^{2}$. Revise the function $\Phi$ in
Corollary 4.3 and follow the same idea to finish the proof.

###  4.2. Rickman’s rugs

A snowflake curve is a segment or a line equipped with a specific metric which
distorts the Euclidean distance by a fixed power less than $1$. More
precisely, $(\mathbb{R},d^{\epsilon})$ is a snowflake curve for any
$\epsilon\in(0,1)$, where $d^{\epsilon}(x,y)=|x-y|^{\epsilon}$ for any
$x,y\in\mathbb{R}$. It is better to introduce the snowflake curve from the
geometric construction.

The _standard snowflake interval_ , i.e., $[0,1]$ equipped with the metric
$d^{\log_{4}3}$, can be constructed by starting with the unit interval
$[0,1]$, then recursively altering each line interval by the following
process:

  1. (1)

divide each line segment into three segments of equal length.

  2. (2)

draw an equilateral triangle that has the middle segment from step
$(\ref{sep})$ as its base and points outward.

  3. (3)

remove the line segment that is the base of the triangle from step
$(\ref{tri})$.

After iterating the above steps $n$ times, we call the resulting curve at the
_$n$ th-stage_. Following the above steps over and over again, the limit curve
is the standard snowflake interval.

Figure 5. Snowflake curve on an interval

A _“flat” snowflake curve_ $\mathcal{L}$, which is more interesting than the
snowflake curve, can be constructed by generalizing the above method. The
construction of $\mathcal{L}$ is still the same as above except with a
revision in step $(\ref{tri})$ at each stage. In step $(\ref{tri})$ of the
$n$th-stage, we don’t construct an equilateral triangle but an isosceles
triangle with the same base where the lengths of legs are $\frac{1}{2}\cdot
l_{n}$ multiple of the length of the base. Furthermore, $l_{n}\to 1$ as
$n\to\infty$. In this way, these triangles on the base segments become more
and more “flat”.

In the following content of this section, we denote by $\mathcal{L}$ the
“flat” snowflake curve constructed on $\mathbb{R}$. It is not hard to see that
all the weak tangents of $\mathcal{L}$ is $\mathbb{R}$. The idea is rather
intuitive. When “blowing up” $\mathcal{L}$ near a point $p$, the “non-
flatness” of $\mathcal{L}$ becomes smaller and smaller. More precisely, for
any $r>0$ and any sequence of balls $\\{B_{i}\\}$ with radius $r$ in the
dilation spaces, the Gromov-Hausdorff distance between $B_{i}$ and a line
segment is smaller and smaller which finally goes to $0$. This finishes the
proof.

Similarly, we can show that all the weak tangents of
$\mathcal{L}\times\mathbb{R}^{n-1}$ are isometric to $\mathbb{R}^{n}$. Here
the metric on $\mathcal{L}\times\mathbb{R}^{n-1}$ is the product metric
$\sqrt{d_{\mathcal{L}}^{2}+|\cdot|^{2}}$.

Rickman observed that, for any $\epsilon\in(0,1)$,
$\mathbb{R}\times\mathbb{R}^{n-1}$ with the product metric
$\sqrt{d^{2\epsilon}+|\cdot|^{2}}$(Rickman’s Rug) is not quasiconformal to
$(\mathbb{R}^{n},|\cdot|)$. For this reason, we are interested in “flat”
snowflake curves.

In [Wu18], Wu constructed the following space which is inspired by the ideas
of Rickman’s rug and the ‘flat” snowflake curves:

Let $\alpha\in(0,1)$, $c\in(0,1)$ and define

$\varphi_{\alpha,c}(x)=\left\\{\begin{array}[]{ll}L(\alpha,c)x,&x\in[0,c];\\\
\left(\frac{x-c(1-\alpha)}{1-c(1-\alpha)}\right)^{\alpha},&x\in[c,1];\end{array}\right.$

where

$L(\alpha,c)=\frac{1}{c}\left(\frac{c\alpha}{1-c(1-\alpha)}\right)^{\alpha}.$

Let $\alpha_{n}$ be an increasing sequence in $(0,1)$ such that
$\lim\limits_{n\to\infty}\alpha_{n}=1$. Then choose $c_{n}\in(0,1)$ such that
$c_{n}\to 0$ and $\lim\limits_{n\to\infty}L(\alpha_{n},c_{n})=+\infty$. We
denote by $\varphi_{n}:=\varphi_{\alpha_{n},c_{n}}$. Choose a sequence $s_{n}$
such that $s_{n}<2(1/n-1/(n+1))$, $s_{n}\cdot L(\alpha_{n},c_{n})$ is
decreasing and $\sum_{n\in\mathbb{N}}s_{n}\cdot L(\alpha_{n},c_{n})<\infty$.

Let $a_{n}=1/n-s_{n}$, $b_{n}=1/n$ and $I_{n}=[a_{n},b_{n}]$. Equipping
$I_{n}$ with the metric

$\delta_{n}=s_{n}\cdot\varphi_{n}\circ\left(\frac{|\ \cdot\ |}{s_{n}}\right).$

The metric $\delta$ on $\mathbb{R}$ is defined in the following way: We assume
that $x<y$, then

$\delta(x,y)=\left\\{\begin{array}[]{ll}\delta_{n}(x,y),&x,y\in I_{n};\\\
|x-y|,&x,y\in\mathbb{R}\setminus\bigcup_{i\in\mathbb{N}}I_{i};\\\
|x-a_{n}|+\delta_{n}(a_{n},y),&x\in\mathbb{R}\setminus\bigcup_{i\in\mathbb{N}}I_{i},y\in
I_{n};\\\ \delta_{n}(x,b_{n})+|b_{n}-y|,&x\in
I_{n},y\in\mathbb{R}\setminus\bigcup_{i\in\mathbb{N}}I_{i};\\\
\delta_{n}(x,b_{n})+|b_{n}-a_{m}|+\delta_{m}(a_{m},y),&x\in I_{n},y\in
I_{m}.\end{array}\right.$

Let

$X_{n}=\left(\mathbb{R}\times\mathbb{R}^{n-1},\sqrt{\delta^{2}+|\ \cdot\
|^{2}}\right).$

Next theorem is the main result of [Wu18]:

######  Theorem 4.6.

Let $n\geq 2$, every weak tangent of $X_{n}$ is isometric to $\mathbb{R}^{n}$
but $X_{n}$ is not quasisymmetric equivalent to $\mathbb{R}^{n}$.

##  5\. Self-quasisymmetricity and quasi-self-symmetricity

In this section we investigate the converse implication, i.e., what are the
consequences if the weak tangents of a metric space $X$ are uniformly
quasisymmetrically embedded into a metric space $Y$?

An appropriate space in this section should illustrate analogous structures
between the local and global scale; thus inspires the following definition.

Let $X$ be a metric space and $p$ be a point in $X$. $X$ is _$\eta$ -self-
quasisymmetric_ at $p$ if there exists a $r_{p}>0$ such that for any
$0<r<r_{p}$, there exists a subset $U\subset B(p,r)$ such that $U$ is
$\eta$-quasisymmetric to $X$. The number $r_{p}$ is called the _standard size_
of $p$.

We say $X$ is _comparable self-quasisymmetric_ at $p$ if there exists a
$C_{p}>0$ such that for any $0<r<r_{p}$, there exists a subset $U$ as above
satisfying $r/C_{p}\leq\mathrm{diam}(U)\leq C_{p}\cdot r$ and $p\in U$. If $X$
is self-quasisymmetric at a set of points, we say it is _uniform comparable_
if $X$ is comparable at these points and the comparable constant $C_{p}$ and
the standard size $r_{p}$ are uniformly bounded.

Any space that is self-quasisymmetric at a point should be bounded since
quasisymmetries preserve boundedness.

Notice that a self-similar space is comparable self-quasisymmetric at any
point in it. See Chapter $2$ of [Hu81] for the definition of self-similarity.

Our intuition tells that self-similar spaces might be a target for a converse
implication. In fact, we can show it works in more generality.

###### Proof of Theorem 1.2.

By definition, we can select a sequence of subsets $\\{U_{n}\\}$, where
$\mathrm{diam}(U_{n})\to 0$, inside sufficiently small open neighborhoods of
$p_{n}$ such that there exists a sequence of $\eta$-quasisymmetric maps

$f_{n}:(U_{n},q_{n},\frac{d_{X}}{\lambda_{n}})\to(X,p,d_{X}).$

We denote by $q_{n}:=f_{n}^{-1}(p)$, and furthermore, $q_{n}\to p$ in $X$.
Notice that $U_{n}$ may not be open or unique.

Let $\lambda_{n}:=\mathrm{diam}(U_{n})$. Without loss of generality, assume
that $\lambda_{n}>0$ for all $n\in\mathbb{N}$ and $\lambda_{n}\to 0$;
otherwise, $X$ is a singleton. We may assume that
$(X,q_{n},\frac{d_{X}}{\lambda_{n}})\xrightarrow{GH}(T_{p}X,p_{\infty},g_{p})$
after passing $\\{\lambda_{n}\\}$ to a suitable subsequence where $T_{p}X$ is
a weak tangent at $p$ . Similarly,
$(U_{n},q_{n},\frac{d_{X}}{\lambda_{n}})\xrightarrow{GH}(U_{p},p_{\infty},g_{p})$
after passing to a suitable subsequence again. It can be directly verified by
Lemma 3.2 that $U_{p}$ is a subset of $T_{p}X$.

It is trivial to see that $(X,p,d_{X})\xrightarrow{GH}(\overline{X},p,d_{X})$.
Notice that $\overline{X}$ is the completion of $X$ since every pointed
Gromov-Hausdorff limit is complete. In the following proof, we will show that
$f_{n}$ subconverges to an $\eta$-quasisymmetry.

|
---|---
$\textstyle{(U_{n},q_{n},\frac{d_{X}}{\lambda_{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{n}}$$\scriptstyle{GH}$$\textstyle{(X,p,d_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{GH}$$\textstyle{(U_{p},p_{\infty},g_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(\overline{X},p,d_{X}).}$

We claim that there exists a sequence of points $x_{n}\in U_{n}$ such that
inequalities (3) hold for $\\{f_{n}\\}$. Take a point $x_{1}$ in $U_{1}$ such
that $\frac{d_{X}}{\lambda_{1}}(q_{1},x_{1})=C$ for some constant $C>0$. We
denote by $x_{n}:=f_{n}^{-1}\circ f_{1}(x_{1})$. Since $f_{1}\circ
f_{n}^{-1}:(U_{n},q_{n},\frac{d_{X}}{\lambda_{n}})\to(U_{1},q_{1},\frac{d_{X}}{\lambda_{1}})$
is an $\eta_{1}(\eta)$-quasisymmetry, by Proposition 2.1, we have

(8)
$\frac{d_{X}({q_{n},x_{n})}}{\mathrm{diam}(U_{n})}\geq\frac{1}{2}\eta_{1}^{-1}\left(\frac{d_{X}(q_{1},x_{1})}{\mathrm{diam}(U_{1})}\right)=\frac{1}{2}\eta_{1}^{-1}\left(C\right).$

Thus

(9) $\frac{1}{2}\eta_{1}^{-1}(C)\leq\frac{d_{X}}{\lambda_{n}}(q_{n},x_{n})\leq
1\qquad\textrm{and}\qquad
d_{X}\left(f_{n}(q_{n}),f_{n}(x_{n})\right)=d_{X}\left(p,f_{1}(x_{1})\right).$

Since
$(U_{n},q_{n},\frac{d_{X}}{\lambda_{n}})\xrightarrow{GH}(U_{p},p_{\infty},g_{p})$
and $(X,p,d_{X})\xrightarrow{GH}(\overline{X},p,d_{X})$, applying Lemma 3.1 to
$\\{f_{n}\\}$ induce an $\eta$-quasisymmetry between $U_{p}$ and $X$. Thus
finishes the proof.

∎

Theorem 1.2 can be further strengthened if adding more conditions.

######  Corollary 5.1.

Let $X$ be a proper, doubling metric space and $p$ be a point in $X$. Assume
that there exists an open neighborhood $V$ of $p$ such that $X$ is uniform
comparable $\eta$-self-quasisymmetric at a dense subset of $V$, then $X$ is
$\eta^{\prime}$-quasisymmetrically embedded into every weak tangent in
$WT_{p}(X)$, where
$\eta^{\prime}(t)=1/\eta^{-1}\\!\\!\left(\frac{1}{t}\right)$.

###### Proof.

Let
$(X,p_{n},\frac{d_{X}}{\lambda_{n}})\xrightarrow{GH}(T_{p}X,p_{\infty},g_{p})$.
Without loss of generality, we assume that $\lambda_{n}$ is smaller than the
standard size and $B(p_{n},\lambda_{n})\subset V$ for all $n\in\mathbb{N}$.
There exists a sequence of subsets $\\{U_{n}\\}$ as above such that
$U_{n}\subset B(p^{\prime}_{n},\frac{\lambda_{n}}{2})\subset
B(p_{n},\lambda_{n})$, $\mathrm{diam}(U_{n})$ is comparable to $\lambda_{n}$
and $p^{\prime}_{n}\in U_{n}$. Here $p^{\prime}_{n}$ is a point at which $X$
is $\eta$-self-quasisymmetric such that $p^{\prime}_{n}$ is sufficiently close
to $p_{n}$. Without loss of generality, we assume that $p_{n}\in U_{n}$,
otherwise, we can construct a $U_{n}$ in the following way. Take a sequence of
points $\\{p^{\prime}_{n_{j}}\\}$ at which $X$ is $\eta$-self-quasisymmetric
such that $\\{p^{\prime}_{n_{j}}\\}$ converges to $p_{n}$, then the
corresponding $U_{n_{j}}$ will (pointed) Hausdorff converges to a subset
$U_{n}$ after passing to a suitable subsequence. Thus $p_{n}\in U_{n}$ and
$U_{n}$ is $\eta$-quasisymmetric to $\overline{X}$. See Corollary 3.4 for a
similar proof.

For every $n\in\mathbb{N}$, there exists an $\eta$-quasisymmetry
$f_{n}:(U_{n},\frac{d_{X}}{\lambda_{n}})\to(\overline{X},d_{X})$.
$(U_{n},p_{n},\frac{d_{X}}{\lambda_{n}})\xrightarrow{GH}(U_{p},p_{\infty},g_{p})\subset(T_{p}X,p_{\infty},g_{p})$
after passing $\\{\lambda_{n}\\}$ to a suitable subsequence. Since
$\overline{X}$ is complete, there exists a sequence of base points
$\\{q_{n}\\}\subset\overline{X}$ where $q_{n}:=f_{n}(p_{n})$ such that
$(\overline{X},q_{n},d_{X})\xrightarrow{GH}(\overline{X},q,d_{X})$ for some
$q\in\overline{X}$ after passing to a suitable subsequence again. Following
the same idea of the proof of Theorem 1.2 and applying Lemma 3.1 on
$\\{f_{n}\\}$ finish the proof. We leave the details to the readers. ∎

######  Corollary 5.2.

Let $X$ be a proper, doubling metric space and $p$ be a point in $X$. Assume
that $X$ is comparable $\eta$-self-quasisymmetric at $p$, then $X$ is
$\eta^{\prime}$-quasisymmetrically embedded into every weak tangent in
$PWT_{p}(X)$, where
$\eta^{\prime}(t)=1/\eta^{-1}\\!\\!\left(\frac{1}{t}\right)$.

The proof of Corollary 5.2 is verbatim the same as the proof of Corollary 5.1.
We omit the proof here.

Self-quasisymmetricity is a rather restrictive property on metric spaces. The
following definition is more practical and could be seen in many other fields.

$X$ is a quasi-self-symmetric space if, roughly speaking, almost every small
piece of $X$ can be uniformly quasisymmetrically mapped back into $X$ with a
bounded size.

Let $X$ be a metric space and $p$ be a point in $X$. $X$ is _quasi-self-
symmetric_ at $p$ if there exist $r_{p}>0,L_{p}\geq 1$ such that for any
$0<r<r_{p}$, there exists an $\eta$-quasisymmetric map $f_{r}$ maps $B(p,r)$
into $X$ such that

(10) $\frac{1}{L_{p}}\leq\mathrm{diam}(f_{r}(B(p,r)))\leq L_{p}.$

The number $r_{p}$ is called the _standard size_ of $p$, the number $L_{p}$ is
called the _bound_ of $p$ and the map $f_{r}$ is called the _distortion map_
of $B(p,r)$. Notice that they may not be unique.

We say a nonempty metric space $X$ is _$\eta$ -quasi-self-symmetric_ if there
exists a dense subset $X_{qss}$ of $X$ such that $X$ is $\eta$-quasi-self-
symmetric at every point in $X_{qss}$ and the standard sizes and the bounds of
these points are uniformly bounded. Thus we can define $r_{0}=\sup_{p\in
X_{qss}}r_{p}$ as the standard size of $X$ and $L=\sup_{p\in X_{qss}}L_{p}$ as
the bound of $X$.

We say a nonempty metric space $X$ is _weakly $\eta$-quasi-self-symmetric_ if
there exists a dense subset $X_{qss}$ of $X$ such that $X$ is $\eta$-quasi-
self-symmetric at every point in $X_{qss}$ and only the bounds of these points
are uniformly bounded.

In the following theorem, we show that quasi-self-symmetricity induces a
converse implication, but not in the global sense.

###### Proof of Theorem 1.3.

We assume that
$(X,p_{n},\frac{d_{X}}{\lambda_{n}})\xrightarrow{GH}(T_{p}X,p_{\infty},g_{p})$.
The proof of this theorem is analogous to the proof of Theorem 1.1 and Theorem
1.2. We split it into two situations: either $X$ is uniformly perfect or not.

Let $X$ be a uniformly perfect metric space. Without loss of generality,
assume that $\lambda_{n}<r_{0}$ for all $n\in\mathbb{N}$. Here $r_{0}$ is the
standard size of $X$. Since $X$ is uniformly perfect, there exists a point
$x_{n}\in\overline{B}(p_{n},\lambda_{n})$ and a constant $C\leq 1$ depending
only on $X$ such that

(11) $C\leq\frac{1}{\lambda_{n}}d_{X}(p_{n},x_{n})\leq 1$

for any $n\in\mathbb{N}$. Let $\\{p^{\prime}_{n}\\}$ be a sequence of points
at which $X$ is $\eta$-quasi-self-symmetric and
$d_{X}(p^{\prime}_{n},p_{n})<C/2\cdot\lambda_{n}$. The reason we select
$p^{\prime}_{n}$ is that $X$ may not be quasi-self-symmetric at $p_{n}$ but
$p_{n}$ can be approximated by such a point.

Since

$\left(\overline{B}(p^{\prime}_{n},\lambda_{n}),p_{n},\frac{d_{X}}{\lambda_{n}}\right)\xrightarrow{GH}(B_{p},p_{\infty},g_{p})$

after passing $\\{\lambda_{n}\\}$ to a suitable subsequence, $B_{p}$ is a
subset of $T_{p}X$ due to Lemma 3.2. Notice that
$\mathrm{diam}(\overline{B}(p^{\prime}_{n},\lambda_{n}))/\lambda_{n}$ may not
be equal to $1$ because there may not exist any point near $\partial
B(p^{\prime}_{n},\lambda_{n})$. However,
$C/2\leq\frac{1}{\lambda_{n}}\mathrm{diam}(B(p^{\prime}_{n},\lambda_{n}))\leq
1$ due to the existence of $x_{n}$.

Since $X$ is compact and quasisymmetries extend to the completions, there
exists a sequence of $\eta$-quasisymmetric maps
$f_{n}:\left(\overline{B}(p^{\prime}_{n},\lambda_{n}),p_{n},\frac{d_{X}}{\lambda_{n}}\right)\to(U_{n},q_{n},d_{X})$
such that $1/L\leq\mathrm{diam}(U_{n})\leq L$, where $U_{n}\subset X$ and $L$
is the bound of $X$. We denote by $q_{n}:=f_{n}(p_{n})$.

Since $X$ and $\\{U_{n}\\}$ are compact, $U_{n}$ Hausdorff converges to a
compact set $U_{q}\subset X$ after passing $\\{U_{n}\\}$ to a suitable
subsequence. Readers may see Theorem $7.3.8$ in [BBI01] for a reference.
Furthermore, we may assume $q_{n_{j}}\to q\in U_{q}$ after passing to a
suitable subsequence again. Notice that (pointed) Hausdorff convergence
implies (pointed) Gromov-Hausdorff convergence. It can be directly verified
that $(U_{n_{j}},q_{n_{j}},d_{X})\xrightarrow{GH}(U_{q},q,d_{X})$.

Thus
$f_{n}:\left(\overline{B}(p^{\prime}_{n},\lambda_{n}),p_{n},\frac{d_{X}}{\lambda_{n}}\right)\to(U_{n},q_{n},d_{X})$
is an $\eta$-quasisymmetry, where

$\left(\overline{B}(p^{\prime}_{n},\lambda_{n}),p_{n},\frac{d_{X}}{\lambda_{n}}\right)\xrightarrow{GH}(B_{p},p_{\infty},g_{p})$

and

$(U_{n},q_{n},d_{X})\xrightarrow{GH}(U_{q},q,d_{X}).$

It remains to show that $f_{n}$ subconverges to an $\eta$-quasisymmetry.

|
---|---
$\textstyle{\left(\overline{B}(p^{\prime}_{n},\lambda_{n}),p_{n},\frac{d_{X}}{\lambda_{n}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{n}}$$\scriptstyle{GH}$$\textstyle{(U_{n},q_{n},d_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{GH}$$\textstyle{(B_{p},p_{\infty},g_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(U_{q},q,d_{X}).}$

Since

$C\leq\frac{1}{\lambda_{n}}d_{X}(p_{n},x_{n})\leq 1,$

it follows Proposition 2.1 and Lemma 3.1 that $\\{f_{n}\\}$ subconverges to an
$\eta$-quasisymmetric map $f^{\prime}:(B_{p},p_{\infty})\to(U_{q},q)$. Since
$B_{p}\subset T_{p}X$, we finish the proof.

If $X$ is not uniformly perfect, the proof is analogous to the second part of
the proof of Theorem 1.1.

Case $1$. If inequality (11) exists, then it follows the above proof.

Case $2$. If not, it is the trivial case.

∎

The following two corollaries follow the same method as Theorem 1.3. We omit
the proofs here.

######  Corollary 5.3.

Let $X$ be a compact, proper, doubling metric space, $p$ be a point in $X$ and
$T_{p}X$ be a weak tangent in $WT_{p}(X)$. Assume that there exists an open
neighborhood $U$ of $p$ such that $X$ is $\eta$-quasi-self-symmetric at a
dense subset of $U$ where the standard sizes and the bounds of these points
are uniformly bounded. Then there exists a ball $B(q,r)$ in $X$ and an
$\eta^{\prime}$-quasisymmetric embedding $f:B(q,r)\to T_{p}X$ such that
$f(q)=p_{\infty}$ and
$\eta^{\prime}(t)=1/\eta^{-1}\\!\\!\left(\frac{1}{t}\right)$. If $X$ is
uniformly perfect, $r$ depends only on $X$ and $\eta$.

######  Corollary 5.4.

Let $X$ be a compact, proper, doubling metric space, $p$ be a point in $X$ and
$T_{p}X$ be a weak tangent in $PWT_{p}(X)$. Assume that there exists a
sequence of points $\\{p_{n}\\}$ such that $X$ is $\eta$-quasi-self-symmetric
at $p_{n}$ for every $n$ and $p_{n}\to p$ in $X$. Furthermore, the standard
sizes and the bounds of $\\{p_{n}\\}$ are uniformly bounded. Then there exists
a ball $B(q,r)$ in $X$ and an $\eta^{\prime}$-quasisymmetric embedding
$f:B(q,r)\to T_{p}X$ such that $f(q)=p_{\infty}$ and
$\eta^{\prime}(t)=1/\eta^{-1}\\!\\!\left(\frac{1}{t}\right)$. If $X$ is
uniformly perfect, $r$ depends only on $X$ and $\eta$.

The following theorem generalizes Theorem 1.3 to weakly quasi-self-symmetric
spaces. It shows that the restriction on the standard size can be weakened.

######  Theorem 5.5.

Let $X$ be a compact, proper, doubling, weakly $\eta$-quasi-self-symmetric
space, $p$ be a point in $X$ and $T_{p}X$ be a weak tangent in $WT_{p}(X)$.
Assume that there exists a neighborhood of $p$ and a constant $C$ depending on
$p$ such that any point $p^{\prime}$, at which $X$ is $\eta$-quasi-self-
symmetric, in this neighborhood should satisfies

$r_{p^{\prime}}\geq C\cdot d_{X}(p,p^{\prime})$

where $r_{p^{\prime}}$ is the standard size of $p^{\prime}$. Then there exists
a ball $B(q,r)$ in $X$ and an $\eta^{\prime}$-quasisymmetric embedding
$f:B(q,r)\to T_{p}X$ such that $f(q)=p_{\infty}$ and
$\eta^{\prime}(t)=1/\eta^{-1}\\!\\!\left(\frac{1}{t}\right)$. If $X$ is
uniformly perfect, $r$ depends only on $X$ and $\eta$.

###### Proof.

The proof is analogous to the proof of Theorem 1.3. The main difference is
that we need to select a suitable sequence of points at which $X$ is quasi-
self-symmetric.

We first prove the case when $X$ is uniformly perfect. Assume that
$(X,p_{n},\frac{d_{X}}{\lambda_{n}})\xrightarrow{GH}(T_{p}X,p_{\infty},g_{p})$.
We select $\\{p^{\prime}_{n}\\}$ in the following way:

  1. (1)

If $d_{X}(p_{n},p)\geq(2+C)\lambda_{n}$, let $p^{\prime}_{n}$ be a point at
which $X$ is $\eta$-quasi-self-symmetric and $d_{X}(p_{n},p^{\prime}_{n})\leq
C\cdot\lambda_{n}$.

  2. (2)

If $d_{X}(p_{n},p)<(2+C)\lambda_{n}$, let $p^{\prime}_{n}$ be be a point at
which $X$ is $\eta$-quasi-self-symmetric and $C_{1}\lambda_{n}\geq
d_{X}(p,p^{\prime}_{n})\geq 2\lambda_{n}$, where $C_{1}$ depends only on $X$.

Thus $r_{p^{\prime}_{n}}\geq C\cdot d_{X}(p,p_{n})\geq 2C\lambda_{n}$ for any
$n\in\mathbb{N}$.

The rest of the proof follows the same idea as Theorem 1.3. We just scratch
the proof below without details.

Let $B(p,r_{0})$ be a ball inside the neighborhood stated in the theorem.
Without loss of generality, assume that $C\lambda_{n}<r_{0}/2$,
$C_{1}\lambda_{n}<r_{0}/2$ and $d_{X}(p_{n},p)<r_{0}/2$ for all
$n\in\mathbb{N}$. We denote by
$B_{n}:=\overline{B}(p^{\prime}_{n},C\lambda_{n})\cup\\{p_{n}\\}$. Thus, by
Lemma 3.2,

$\left(B_{n},p_{n},\frac{d_{X}}{\lambda_{n}}\right)\xrightarrow{GH}(B_{p},p_{\infty},g_{p})\subset
T_{p}X$

after passing $\\{\lambda_{n}\\}$ to a suitable subsequence. Furthermore,
$\overline{B}(p^{\prime}_{n},C\lambda_{n})$ is $\eta$-quasisymmetric to
$U^{\prime}_{n}$, where $U^{\prime}_{n}\subset X$,
$\frac{1}{L}\leq\mathrm{diam}(U^{\prime}_{n})\leq L$ and $L$ is the bound of
$X$.

Let $q_{n}$ be a point in $X$ such that there exists a constant $C_{2}$
depending on $X$ where $d_{X}(q_{n},x)>C_{2}$ for every $x\in U^{\prime}_{n}$.
We claim that such a $q_{n}$ exists, otherwise adjust
$\overline{B}(p^{\prime}_{n},C\lambda_{n})$ to get a smaller $U_{n}$. After
adding $q_{n}$ as a base point of $U^{\prime}_{n}$ if needed, we denote by

$U_{n}:=\left\\{\begin{array}[]{ll}U_{n}&\textrm{if}\
p_{n}\in\overline{B}(p^{\prime}_{n},C\lambda_{n});\\\ U_{n}\cup
q_{n}&\textrm{if}\
p_{n}\notin\overline{B}(p^{\prime}_{n},C\lambda_{n}).\end{array}\right.$

It can be directly verified that $B_{n}$ is quasisymmetric to $U_{n}$, since
adding a point selected as above doesn’t affect the quasisymmetry between
bounded spaces. Assume that $U_{n}$ is pointed Hausdorff converges to $U_{q}$
after passing to a suitable subsequence. Thus $B_{p}$ is quasisymmetric to
$U_{q}$ by Lemma 3.1 and we finish the proof.

If $X$ is not uniformly perfect, we follow the same idea as Theorem 1.3 and
finish the proof.

∎

Let $X$ be a quasi-self-symmetric space. We say $X$ satisfies the covering
property if for any collection of sufficiently small balls that are centered
at every point of $X_{qss}$, there exists a collection of distortion maps
whose image covers $X$. More precisely, let $X$ be a quasi-self-symmetric
space and $r_{1}<r_{0}$ be some constant. We let $\\{B(x_{i},r_{i}):x_{i}\in
X_{qss},0<r_{i}<r_{1}\\}$ be a collection of balls which centered at every
point of $X_{qss}$ with radius less than $r_{1}$ for every $i$. $X$ satisfies
the _covering property_ if for every such a collection, there exists a
collection of distortion maps $\\{f_{i}\\}$ such that
$\\{f_{i}(B(x_{i},r_{i}))\\}$ covers $X$.

The covering property is a restriction on quasi-self-symmetric spaces. We are
interested in a metric space which is locally “embeddable” everywhere. If
there is no covering property, only part of the space is interesting, namely,
the union of the images under all distortion maps, but the rest is not
explorable. Many interesting examples satisfy the covering property and it is
not clear that if there exists a quasi-self-symmetric space without it.

Applying the covering property to a quasi-self-symmetric space induce the the
following corollary:

######  Corollary 5.6.

Let $X$ be a compact, proper, doubling, $\eta$-quasi-self-symmetric space with
the covering property. Assume that for every point $p\in X$, there exists one
weak tangent in $WT_{p}(X)$ that can be $\theta$-quasisymmetrically embedded
into a metric space $Y$, then there exists a constant $r^{\prime}>0$ such that
every ball $B(x,r)$ in $X$ with radius less than $r^{\prime}$ can be
$\theta^{\prime}$-quasisymmetrically embedded into $Y$ where $\theta^{\prime}$
depends on $\theta$ and $\eta$. More precisely,
$\theta^{\prime}=\theta\circ\eta^{\prime}$ where
$\eta^{\prime}(t)=1/\eta^{-1}\\!\\!\left(\frac{1}{t}\right)$.

Corollary 5.6 is a direct result of Theorem 1.3 and the covering property.

##  6\. Expanding dynamics on metric spaces

In this section, we investigate metric spaces with expanding dynamics and
apply our results to them.

###  6.1. Expanding maps

Let $X$ be a metric space. A map $f:X\to X$ is called _expanding_ if for every
point $x\in X$, there exists a neighborhood $U$ of $x$ and a constant $L>1$
such that

$d_{X}(f(x_{1}),f(x_{2}))\geq L\cdot d_{X}(x_{1},x_{2})$

for any $x_{1},x_{2}\in U$. Intuitively, expanding maps locally expand the
distance of a pair of points.

For example, an expanding endomorphism defined on a compact Riemannian
manifold $M$ is a smooth function $f:M\to M$ such that there exists a $C>1$
such that $||Tf(v)||\geq C\cdot||v||$ for all $v\in TM$. See [Sh69] for a
general exploration of expanding endomorphism on Riemannian manifolds.

However, equipping with an expanding map is not sufficient to illustrate
interesting structures(e.g., self-similar) on a metric space. Roughly
speaking, the reason is that the local structures of the expanding images of
two points maybe much different after sufficiently large iterations; thus it
is essential to require a control of the growth of expansion.

If $f$ is an expanding map, we say $f$ is _controlled_ if there exists
$\alpha\geq 1,C>0$ such that

$\frac{d_{X}(f(x),f(y))}{d_{X}(x,y)}-\frac{d_{X}(f(x),f(z))}{d_{X}(x,z)}\leq
C\cdot\mathrm{diam}(\\{x,y,z\\})^{\alpha}$

for any $x,y,z\in U$. Intuitively, it is the hölder continuity of the “first
derivative”, which controls the growth of expansion.

Assume that $f$ is an expanding endomorphism on a compact Riemannian manifold
$M$. In this case, compactness and differentiability together bound the growth
of expansion.

A metric space $X$ is called _$H$ -quasi-self-similar_ if there exist
$r_{0}>0,H\geq 1$ such that given any ball $B$ with radius $r<r_{0}$, there
exists a $H$-bilipschitz map $f_{B}$ which maps $B$ into $X$ such that

(12) $\frac{1}{H}\frac{r_{0}}{r}d_{X}(x,y)\leq d_{X}(f_{B}(x),f_{B}(y))\leq
H\frac{r_{0}}{r}d_{X}(x,y)$

for all $x,y\in B$. In fact, it is a special case of quasi-self-symmetric.

Equipping controlled expanding dynamics on a metric space generates quasi-
self-similarity. It follows the same idea of the distortion lemma on P.$42$ of
[Su82]. For the sake of completeness, we scratch a proof of it.

######  Theorem 6.1.

Let $X$ be a compact metric space and $\\{U_{i}\\}_{i=1}^{N}$ be a finite open
cover of $X$. If for each $i$, there exists $f_{i}:U_{i}\to X$, $L_{i}>1$,
$\alpha_{i}\geq 1$, and $C_{i}>0$ such that

  1. (1)

$d_{X}(f_{i}(x),f_{i}(y))\geq L_{i}\cdot d_{X}(x,y)$;

  2. (2)

$\frac{d_{X}(f_{i}(x),f_{i}(y))}{d_{X}(x,y)}-\frac{d_{X}(f_{i}(x),f_{i}(z))}{d_{X}(x,z)}\leq
C_{i}\cdot\mathrm{diam}(\\{x,y,z\\})^{\alpha_{i}}$;

for every $x,y,z\in U_{i}$, then $X$ is quasi-self-similar.

###### Proof.

Let $r_{0}$ be the Lebesgue number of $\\{U_{i}\\}_{i=1}^{N}$. For any
$B(x,r)$ with $r<r_{0}$, $B(x,r)$ should be a subset of one of
$\\{U_{i}\\}_{i=1}^{N}$. Applying the corresponding $f_{i}$ to $B(x,r)$ maps
it to a larger image. Iterating this action until we get an image whose
diameter is no smaller than $r_{0}$.

Suppose that $B(x,r)=B_{i_{0}}\subset U_{i_{0}}$, and after iterating $n$
times we expand $B_{i_{0}}$ to $B_{i_{n}}\subset U_{i_{n}}$. It is sufficient
to prove that there exists a constant $H>1$ independent of $n$ such that

$\frac{1}{H}\cdot\frac{d_{X}(x^{\prime}_{n},y^{\prime}_{n})}{d_{X}(x^{\prime},y^{\prime})}\leq\frac{d_{X}(x_{n},y_{n})}{d_{X}(x,y)}\leq
H\cdot\frac{d_{X}(x^{\prime}_{n},y^{\prime}_{n})}{d_{X}(x^{\prime},y^{\prime})}.$

where $x,y,x^{\prime},y^{\prime}\in B_{i_{0}}$ and
$x_{j},y_{j},x^{\prime}_{j},y^{\prime}_{j}$ are the corresponding images of
them in $B_{i_{j}},j=1,\ldots,n$. We denote by
$D_{j}:=\mathrm{diam}(\\{x_{j},y_{j},x^{\prime}_{j},y^{\prime}_{j}\\})$.

By conditions (1) and (2), we have

$\left(\frac{d_{X}(f_{i}(x),f_{i}(y))}{d_{X}(x,y)}-\frac{d_{X}(f_{i}(x),f_{i}(z))}{d_{X}(x,z)}\right)/\left(\frac{d_{X}(f_{i}(x),f_{i}(z))}{d_{X}(x,z)}\right)\leq\frac{C_{i}}{L_{i}}\cdot\mathrm{diam}(\\{x,y,z\\})^{\alpha_{i}},$

which implies

$\displaystyle\frac{d_{X}(f_{i}(x),f_{i}(y))}{d_{X}(x,y)}/\frac{d_{X}(f_{i}(x),f_{i}(z))}{d_{X}(x,z)}$
$\displaystyle\leq$ $\displaystyle
1+\frac{C_{i}}{L_{i}}\cdot\mathrm{diam}(\\{x,y,z\\})^{\alpha_{i}}$
$\displaystyle\leq$ $\displaystyle e^{\lambda\cdot\mathrm{diam}(\\{x,y,z\\})}$

for any $x,y,z\in U_{i}$ and for any $i$. Here $\lambda$ is a constant which
depends on $\\{L_{i}\\},\\{\alpha_{i}\\}$ and $\\{C_{i}\\}$. Thus

$\frac{d_{X}(f_{i}(x),f_{i}(y))}{d_{X}(x,y)}/\frac{d_{X}(f_{i}(z),f_{i}(w))}{d_{X}(z,w)}\leq
e^{\lambda^{\prime}\cdot\mathrm{diam}(\\{x,y,z,w\\})}$

for any $x,y,z,w\in U_{i}$ and for any $i$, where $\lambda^{\prime}$ depends
only on $\lambda$.

$\displaystyle\log\left(\frac{d_{X}(x_{n},y_{n})}{d_{X}(x,y)}/\frac{d_{X}(x^{\prime}_{n},y^{\prime}_{n})}{d_{X}(x^{\prime},y^{\prime})}\right)$
$\displaystyle=$ $\displaystyle\
\sum_{j=1}^{n}\log\left(\left(\frac{d_{X}(x_{j},y_{j})}{d_{X}(x_{j-1},y_{j-1})}\right)/\left(\frac{d_{X}(x^{\prime}_{j},y^{\prime}_{j})}{d_{X}(x^{\prime}_{j-1},y^{\prime}_{j-1})}\right)\right)$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{n}\log\left(e^{\lambda^{\prime}D_{j-1}}\right)$
$\displaystyle\leq$ $\displaystyle\lambda^{\prime}\sum_{j=1}^{n}D_{j-1}.$

Since $\\{D_{j}\\}$ is bounded by a geometric series, this finishes the proof.
∎

The collection $\\{U_{i}\\}_{i=1}^{N}$ in Theorem 6.1 is called a _controlled
expanding cover_ of $X$. If $X$ is a compact metric space and $f:X\to X$ is a
controlled expanding map, then $f$ induces a finite controlled expanding cover
on $X$; thus $X$ is quasi-self-similar.

###  6.2. Gromov hyperbolic spaces and groups

A length space $(X,d)$ is called _$\delta$ -hyperbolic_ (where $\delta\geq 0$)
if for any triangle with geodesic sides in $X$, each side of the triangle is
contained in the $\delta$-neighborhood of the union of two other sides. A
finitely generated group $G$ is called _hyperbolic_ if there exists a
symmetric finite generating set $S$ for $G$ and a positive constant $\delta$
such that the Cayley graph $\Gamma(G,S)$ of $G$ with respect to $S$ is
$\delta$-hyperbolic. A group $G$ is equipped with the _word metric_ with
respect to $S$ is equivalent to its Cayley graph $\Gamma(G,S)$ is equipped
with the intrinsic metric. See [BH99], [KB02] for a reference of Gromov
hyperbolic spaces and groups.

Let $(X,d)$ be a metric space and $x,y,p\in X$. The _Gromov product_ of $x,y$
with respect to $p$ is defined by

$(x,y)_{p}=\frac{1}{2}\left(d(x,p)+d(y,p)-d(x,y)\right).$

In hyperbolic metric spaces, Gromov product measures how long two geodesics
travel close together. More precisely, if $x,y,p$ are three distinct points in
a $\delta$-hyperbolic space $X$, then the initial segments of length
$(x,y)_{p}$ of any two geodesics connecting $x,p$ and $y,p$ are
$2\delta$-Hausdorff close.

Let $(X,d)$ be a proper $\delta$-hyperbolic metric space and $p$ be a chosen
base point of $X$. We define the _boundary at infinity_ of $X$ by

$\partial_{\infty}X:=\\{[r]:r:[0,\infty)\to X\ \textrm{is a geodesic ray},\
r(0)=p\\},$

where two geodesic rays $r_{1},r_{2}$ are equivalent if
$d_{H}(r_{1},r_{2})<\infty$.

Equivalently, we also have

$\partial_{\infty}X:=\\{[\\{x_{n}\\}]:\\{x_{n}\\}_{n=1}^{\infty}\ \textrm{is a
sequence converging to infinity in}\ X\\},$

where $\\{x_{n}\\}$ converges to infinity if
$\liminf\limits_{i,j\to\infty}(x_{i},x_{j})_{p}=\infty$, and
$\\{x_{n}\\},\\{y_{n}\\}$ are equivalent if
$\liminf\limits_{i,j\to\infty}(x_{i},y_{j})_{p}=\infty$. It can be directly
verified that the definitions do not depend on the base point.

We extend the Gromov product to $X\cup\partial_{\infty}X$ by

$(x,y)_{p}=\sup\liminf_{i,j\to\infty}(x_{i},y_{j})_{p}$

where the supremum is taken over all sequences $\\{x_{i}\\}$ and $\\{y_{j}\\}$
in $X$ such that $x=[\\{x_{i}\\}]$ and $y=[\\{y_{i}\\}]$.

The following properties are important in studying the boundaries at infinity
of hyperbolic spaces. See Remarks $3.17$ in Chapter III.H of [BH99].

######  Proposition 6.2.

Let $X$ be a $\delta$-hyperbolic space and $p$ be a chosen base point.

  1. (1)

For any $x,y,z\in X\cup\partial_{\infty}X$, we have

(13) $(x,y)_{p}\geq\min\\{(x,z)_{p},(y,z)_{p}\\}-2\delta.$

  2. (2)

For any $x,y\in\partial_{\infty}X$ with $x=[\\{x_{i}\\}]$ and
$y=[\\{y_{i}\\}]$, we have

(14)
$(x,y)_{p}-2\delta\leq\liminf_{i,j\to\infty}(x_{i},y_{j})_{p}\leq(x,y)_{p}.$

There is a natural topology on $\partial_{\infty}X$, where the basis of this
topology is the collection of

$V(x,\varepsilon):=\\{y\in\partial_{\infty}X:\exists\ \textrm{gedeosic rays}\
r,r^{\prime}\ \textrm{starting at}\ p\\\ \textrm{such that}\
[r]=x,[r^{\prime}]=y\ \textrm{and}\
\liminf_{t\to\infty}(r(t),r^{\prime}(t))_{p}\geq\varepsilon\\}$

for every $x\in X$ and $r>0$.

Equivalently, we also have

$V(x,\varepsilon):=\\{y\in\partial_{\infty}X:\exists\ \textrm{sequences}\
\\{x_{n}\\},\\{y_{n}\\}\\\ \textrm{such that}\
[\\{x_{n}\\}]=x,[\\{y_{n}\\}]=y\ \textrm{and}\
\liminf_{i,j\to\infty}(x_{i},y_{j})_{p}\geq\varepsilon\\}$

for every $x\in X$ and $r>0$. Moreover, $\partial_{\infty}X$ is compact with
this topology.

We say a metric $d_{a}$ on $\partial_{\infty}X$ is a _visual metric_ with
respect to the base point $p$ and the visual parameter $a>1$ if there is
$C_{1},C_{2}>0$ such that the following holds:

  1. (1)

The metric $d_{a}$ induces the natural topology on $\partial_{\infty}X$;

  2. (2)

For any two distinct points $x,y\in\partial_{\infty}X$,

(15) $C_{1}a^{-(x,y)_{p}}\leq d_{a}(x,y)\leq C_{2}a^{-(x,y)_{p}}.$

We say two metric spaces $X$ and $Y$ are _quasi-isometric_ if there exists a
map $f:X\to Y$ and $C_{1}\geq 1,C_{2}\geq 0$ such that

  1. (1)

For any two points $x_{1},x_{2}\in X$,

(16) $\frac{1}{C_{1}}d_{X}(x_{1},x_{2})-C_{2}\leq d_{Y}(f(x_{1}),f(x_{2}))\leq
C_{1}d_{X}(x_{1},x_{2})+C_{2}.$

  2. (2)

For any $y\in Y$, there exists a $x\in X$ such that $d_{Y}(f(x),y)\leq C_{2}$.

The following propositions illustrate several important properties of
hyperbolic spaces.

######  Proposition 6.3.

Let $X$ be a proper $\delta$-hyperbolic space, then

  1. (1)

There exists a $a_{0}>1$ such that for any base point $p\in X$ and any
$a\in(1,a_{0})$, the boundary $\partial_{\infty}X$ admits a visual metric
$d_{a}$ with respect to $p$ and $a$.

  2. (2)

Suppose $d^{\prime}$ and $d^{\prime\prime}$ are visual metrics on
$\partial_{\infty}X$ with respect to the same visual parameter $a$ and the
base points $p^{\prime}$ and $p^{\prime\prime}$, respectively. Then
$d^{\prime}$ and $d^{\prime\prime}$ are bi-Lipschitz equivalent.

  3. (3)

Suppose $d^{\prime}$ and $d^{\prime\prime}$ are visual metrics on
$\partial_{\infty}X$ with respect to the visual parameters $a^{\prime}$ and
$a^{\prime\prime}$ and the base points $p^{\prime}$and $p^{\prime\prime}$,
respectively. Then $d^{\prime}$ and $d^{\prime\prime}$ are Hölder-equivalent.

######  Proposition 6.4.

Let $X$ and $Y$ be proper hyperbolic spaces and $f:X\to Y$ is a quasi-
isometry, then $f$ extends to a quasisymmetric map
$\hat{f}:\partial_{\infty}X\to\partial_{\infty}Y$, where the boundaries at
infinity are equipped with visual metrics.

We define the _boundary at infinity of a hyperbolic group $G$_ by
$\partial_{\infty}G:=\partial_{\infty}\Gamma(G,S)$. A group $G$ acts on a
length space $X$ _geometrically_ if the action is isometric, i.e., every $g\in
G$ acts on $X$ as an isometry, cocompact, i.e., $X/G$ is compact, and properly
discontinuous, i.e., for any compact $K\subset X$ the set $\\{g\in G:gK\cap
K=\emptyset\\}$ is finite. For example, a finitely generated group $G$ acts on
$\Gamma(G,S)$ geometrically for any symmetric finite generating set $S$.

######  Theorem 6.5 (Švarc-Minor).

Let $G$ be a group act geometrically on a length space $X$. Then the group $G$
is finitely generated, the space $X$ is proper and for any symmetric finite
generating set $S$ of G, there exists a quasi-isometry between $\Gamma(G,S)$
and $X$.

Notice that every element $g$ of a hyperbolic group $G$ induces a
quasisymmetric automorphism $\hat{g}$ on $\partial_{\infty}G$.

The following theorem shows the existence of an expanding cover on
$\partial_{\infty}G$.

######  Theorem 6.6.

Let $G$ be a hyperbolic group and $\partial_{\infty}G$ be its boundary at
infinity. For any visual metric $d$ on $\partial_{\infty}G$, there exists a
finite expanding cover on $(\partial_{\infty}G,d)$.

###### Proof.

We claim that the standard translation action of a hyperbolic group G on its
boundary at infinity $\partial_{\infty}G$, equipped with visual metric $d$, is
expanding. Namely, for any point $p\in\partial_{\infty}G$ there exist a
neighborhood $U$ of $p$, $g\in G$ and $\alpha>1$ such that for every $x,y\in
U$, $d(gx,gy)\geq\alpha d(x,y)$.

Without loss of generality, we assume that in this proof the visual metric on
$\partial_{\infty}G$ is with respect to $1\in G$; otherwise, it will be
sufficient to substitute $g$ with $gh^{-1}$ where $h$ is the original base
point.

We first prove the theorem when $G$ is a free group
$G:=F(a_{1},\ldots,a_{r})$.

Let $x,y$ be reduced words(i.e., the simplest representation in generators) on
$G\cup\partial_{\infty}G$, and let the Gromov product of them with respect to
the identity element $1$ be $(x,y)_{1}=t$. Thus $t$ is the length of the
maximal common initial segment of $x,y$.

For any visual metric $d$ on $\partial_{\infty}G$, there exist
$C_{1},C_{2}>0,a>1$ such that

$C_{1}\cdot a^{-(x,y)_{1}}\leq d(x,y)\leq C_{2}\cdot a^{-(x,y)_{1}}$

for every $x,y\in\partial_{\infty}G$. Notice that $a\in(1,\infty)$ since the
Cayley graph of $G$ is a tree.

Let $p$ be a point in $\partial_{\infty}G$. We think of $p$ as a semi-
infinite(bounded in one direction) reduced word in $F(a_{1},\ldots,a_{r})$.
For any integer $m\geq 0$, we denote by

$U(p,m):=\\{p^{\prime}\in\partial_{\infty}G:(p,p^{\prime})_{1}\geq m\\}.$

This is well defined since every element in $\partial_{\infty}G$ has a unique
representation in the chosen generators.

Take any $p\in\partial_{\infty}G$ and let $m\geq 1$ be sufficiently large
(where sufficiently large depends on $a$, $C_{1}$ and $C_{2}$). Let $w\in
F(a_{1},\ldots,a_{r})$ be the element given by the initial segment of $p$ of
length $m$ and let $g=w^{-1}$. We claim that for any $x,y\in U(p,m)$,
$d(gx,gy)\geq\alpha d(x,y)$ for some $\alpha>1$.

Let $x,y\in U(p,m)$. Then, viewed as semi-infinite reduced word in $G$, both
$x$ and $y$ have $w$ as their initial segment of length $m$, so
$x=wx^{\prime}$ and $y=wy^{\prime}$ where $x^{\prime},y^{\prime}$ are semi-
infinite reduced words and thus elements of $\partial_{\infty}G$.

We have $gx=w^{-1}wx^{\prime}=x^{\prime}$ and
$gy=w^{-1}wy^{\prime}=y^{\prime}$. Let $v$ be the maximal common initial
segment of $x^{\prime},y^{\prime}$, then $wv$ is the maximal common initial
segment of $x,y$. Let $|v|$ denotes the length of $v$ under word metric; thus
we have $(x^{\prime},y^{\prime})_{1}=|v|$ and $(x,y)_{1}=|v|+m$.

Hence

$d(gx,gy)=d(x^{\prime},y^{\prime})\geq C_{1}\cdot\frac{1}{a^{|v|}}$

and

$d(x,y)\leq C_{2}\cdot\frac{1}{a^{|v|+m}}.$

Thus

$d(gx,gy)\geq\frac{C_{1}}{C_{2}}a^{m}d(x,y).$

Since $m$ is sufficiently large, we define $\alpha:=C_{1}/C_{2}\cdot a^{m}>1$.
This finishes the proof.

If $G$ is an arbitrary hyperbolic group, the argument is similar. Let
$\Gamma(G,S)$ be a Cayley graph of $G$ and $d$ be any visual metric on
$\partial_{\infty}G=\partial_{\infty}\Gamma(G,S)$ with respect to the base
point $1$ and the visual parameter $a$. We assume that $\Gamma(G,S)$ is
$\delta$-hyperbolic. There exist $C_{1},C_{2}>0$ such that for any
$x,y\in\partial_{\infty}G$,

$C_{1}\cdot a^{-(x,y)_{1}}\leq d(x,y)\leq C_{2}\cdot a^{-(x,y)_{1}}.$

For any $m\geq 0$ and $p\in\partial_{\infty}G$, we define

$U(p,m):=\\{q\in\partial_{\infty}G:\exists\ \textrm{geodesic rays}\
r,r^{\prime}\ \textrm{starting at}\ 1\\\ \textrm{such that}\
[r]=p,[r^{\prime}]=q\ \textrm{and}\
\liminf_{t\to\infty}(r(t),r^{\prime}(t))_{1}\geq 2m\\}.$

Notice that $U(p,m)$ is a neighborhood of $p$ in $\partial_{\infty}G$ (in
fact, it is a basic neighborhood in the original definition of the topology on
$\partial_{\infty}G$).

Take any $p\in\partial_{\infty}G$ and let $m\geq 1$ be sufficiently large
(where sufficiently large depends on $a$, $C_{1}$, $C_{2}$ and the
hyperbolicity constant $\delta$). Let $r_{p}$ be a geodesic ray representing
$p$ and $w\in G$ be the element given by the initial segment of $r_{p}$ of
length $m$. Let $g:=w^{-1}$. We claim that for any $x,y\in U(p,m)$,
$d(gx,gy)\geq\alpha d(x,y)$ for some $\alpha>1$.

Notice that for any other geodesic ray $r_{p}^{\prime}$ representing $p$, we
have $d_{H}(r_{p},r_{p}^{\prime})<\infty$. Thus
$\liminf_{t\to\infty}(r_{p}(t),r_{p}^{\prime}(t))_{1}=\infty$. Since $m$ is
sufficiently large, we may assume that, by Proposition 6.2, there exist two
geodesic rays $r_{x}$ and $r_{y}$ representing $x$ and $y$ in $U(p,m)$,
respectively, such that

$\liminf_{t\to\infty}(r_{x}(t),r_{p}(t))_{1}\geq
m\qquad\textrm{and}\qquad\liminf_{t\to\infty}(r_{y}(t),r_{p}(t))_{1}\geq m.$

Let $d_{G}$ denotes the word metric on $G$ with respect to $S$. Then

$\displaystyle(r_{x}(t),r_{y}(t))_{1}-(gr_{x}(t),gr_{y}(t))_{1}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(d_{G}(r_{x}(t),1)-d_{G}(gr_{x}(t),1)+d_{G}(gr_{y}(t),1)-d_{G}(r_{y}(t),1)\right).$

for every $t$.

Let $t$ be sufficiently large, i.e., $d_{G}(r_{x}(t),1)\geq m$ and
$d_{G}(r_{y}(t),1)\geq m$. Notice that, by the definition of Gromov product,
there exists a number $t^{\prime}$ where $d_{G}(r_{x}(t^{\prime}),1)\leq m$
such that $d_{G}(r_{x}(t^{\prime}),w)\leq 2\delta$. Thus, by analyzing the
geodesic triangles, for sufficiently large $t$ we have

(17)
$d_{G}(r_{x}(t),1)-d_{G}(gr_{x}(t),1)=d_{G}(r_{x}(t),1)-d_{G}(r_{x}(t),w)\geq
m-4\delta.$

Figure 6. Geodesic triangles involve $1$, $w$, $r(t)$

Similarly, for sufficiently large $t$ we also have

(18)
$d_{G}(r_{y}(t),1)-d_{G}(gr_{y}(t),1)=d_{G}(r_{y}(t),1)-d_{G}(r_{y}(t),w)\geq
m-4\delta.$

This finally shows that

(19) $(r_{x}(t),r_{y}(t))_{1}-(gr_{x}(t),gr_{y}(t))_{1}\geq m-4\delta$

for sufficiently large $t$.

By Proposition 6.2, we have

(20) $|(x,y)_{1}-\liminf_{t\to\infty}(r_{x}(t),r_{y}(t))_{1}|\leq 2\delta.$

Similarly, $gr_{x},gr_{y}$ are two geodesic rays representing $gx$ and $gy$ on
$\partial_{\infty}G$, respectively. Thus

(21) $|(gx,gy)_{1}-\liminf_{t\to\infty}(gr_{x}(t),gr_{y}(t))_{1}|\leq
2\delta.$

Combining inequalities (19) with (20) and (21), we have

$(x,y)_{1}-(gx,gy)_{1}\geq m-8\delta.$

Namely, $(x,y)_{1}$ equals $m+(gx,gy)_{1}$ up to an additive error of
$8\delta$.

Hence

$d(gx,gy)\geq C_{1}\cdot a^{-(gx,gy)_{1}}\geq C_{1}\cdot
a^{m-8\delta}a^{-(x,y)_{1}}\geq\frac{C_{1}}{C_{2}}\cdot a^{m-8\delta}d(x,y)$

Since $m$ is sufficiently large, we define $\alpha:=C_{1}/C_{2}\cdot
a^{m-8\delta}>1$. This finishes the proof.

∎

Theorem 6.6 is also valid in a more general situation. Let $X$ be a proper
hyperbolic metric space and $p$ be a chosen base point. Let $G$ be a group
acting on $X$ by isometries. We say $\Lambda(G)$ is the _limit set_ of $G$ on
$X$ if

$\Lambda(G):=\\{[\\{g_{n}p\\}]\in\partial_{\infty}X:\exists\ g_{n}\in G\
\textrm{such that}\\\ \\{g_{n}p\\}_{n=1}^{\infty}\ \textrm{is a sequence
converging to infinity in}\ X\\}.$

This definition is still independent of the base point. One can also define
the _conical limit set_ $\Lambda_{c}(G)$ as the collection of points
$q\in\Lambda(G)$ which are approximated by a sequence from the orbit $Gp$ such
that this sequence is contained in a bounded neighborhood of some geodesic ray
$r$ with $[r]=q$. Notice that $\partial_{\infty}G$ is a conical limit set of
$G$. In this situation, we also get a corresponding translation action of $G$
on $\Lambda_{c}(G)$.

######  Corollary 6.7.

Let $(X,d)$ be a proper hyperbolic metric space, $G$ be a group acting on $X$
by isometries and $\Lambda_{c}(G)$ be the conical limit set of $G$ on $X$. For
any visual metric $d$ on $\partial_{\infty}X$, there exists a finite expanding
cover on $\Lambda_{c}(G)$.

The proof of Corollary 6.7 is literally the same as the proof of Theorem 6.6.
We leave it to the readers.

$\Lambda_{c}(G)$ admits a finite expanding cover; however, it is not
sufficient to generate quasi-self-similarity. In other words, we need to
control the expansion.

In the following content of this section, we always assume that the boundary
at infinity of a hyperbolic space contains more than two points(i.e., non-
elementary).

Let $G$ be a group and $H$ be a subgroup of $G$ of finite index. Then $H$ is
hyperbolic if and only if $G$ is hyperbolic and in that case
$\partial_{\infty}H=\partial_{\infty}G$. A group $G$ is _virtually free_ if
there exists a finite index subgroup of $G$ that is a free group.

######  Corollary 6.8.

Let $G$ be a virtually free group and $\partial_{\infty}G$ be its boundary at
infinity. There exists a visual metric $d$ on $\partial_{\infty}G$ such that
$(\partial_{\infty}G,d)$ is quasi-self-similar.

###### Proof.

It is sufficient to prove the corollary for free groups. We use the same
notation as Theorem 6.6. Let $d(x,y)=2^{-(x,y)_{1}}$ and for any
$p\in\partial_{\infty}G$, take the neighborhood $U(p,1)$. Let $w$ be the
element given by the initial segment of $p$ of length $1$ and let $g=w^{-1}$.
It can be directly verified that $d$ is a visual metric since the Cayley graph
of $G$ is a tree.

It follows the proof of Theorem 6.6 that $d(gx,gy)=2d(x,y)$ for any $x,y\in
U(p,1)$. Thus $g$ acts on $U(p,1)$ to generate a controlled expanding map. We
finish the proof by Theorem 6.1. ∎

The boundary at infinity of a virtually free group is not only quasi-self-
similar, but it also obtains some kind of self-similar property(not the same
as in [Hu81]).

If $X$ is any compact metrizable space that has at least three points, we
denote the space of distinct triples of $X$ by $\Sigma_{3}(X)$. Namely,

$\Sigma_{3}(X)=\\{(o,p,q)\in X^{3}:o\neq p,o\neq q,p\neq q\\}.$

Assume that a group $G$ acts on $X$ by homeomorphisms. Such an action induces
a diagonal action of $G$ on $\Sigma_{3}(X)$. If the action of $G$ on
$\Sigma_{3}(X)$ is properly discontinuous and cocompact, we say $G$ acts on
$X$ as a _uniform convergence group_.

Let $G$ be a group acting on a proper hyperbolic metric space by isometries.
If $G$ acts on $\Lambda(G)$ as a uniform convergence group, then $G$ is a
hyperbolic group and $\Lambda(G)=\Lambda_{c}(G)$. Readers may see Section $5$
of [KB02] for a reference.

$\Gamma$ is a _Kleinian group_ if $\Gamma$ is a discrete subgroup of
isometries of hyperbolic space $\mathbb{H}^{n},n\geq 3$. Let
$\textrm{Hull}(\Lambda(\Gamma))\subset\mathbb{H}^{n}$ denotes the _convex
hull_ of the limit set, i.e., the smallest convex set containing all geodesics
with both endpoints in $\Lambda(\Gamma)$. Then $\Gamma$ acts _convex
cocompactly_ on $\mathbb{H}^{n}$ if $\textrm{Hull}(\Lambda(\Gamma))/\Gamma$ is
compact. If $\Gamma$ is a convex cocompact Kleinian group, then $\Gamma$ acts
on $\Lambda(\Gamma)$ as a uniform convergence group.

######  Corollary 6.9.

Let $\Gamma$ be a Kleinian group and let $\Gamma$ acts on $\Lambda(\Gamma)$ as
a uniform convergence group. Then $\Lambda(\Gamma)$, equipped with spherical
metric, is quasi-self-similar.

###### Proof.

Since $\Gamma$ acts on $\Lambda(\Gamma)$ as a uniform convergence group,
$\Lambda(\Gamma)=\Lambda_{c}(\Gamma)$. We can naturally identify
$\partial_{\infty}\mathbb{H}^{n}$ as $\mathbb{S}^{n-1}$ and equip it with the
spherical metric(as a visual metric).

Notice that every $g\in\Gamma$ acting on $\Lambda(\Gamma)$ induces a conformal
map(a restriction of a Möbius transformation). Since $\Lambda(\Gamma)$ is
compact, $g^{\prime}$ and $g^{\prime\prime}$ are bounded for any $g\in\Gamma$.
By Corollary 6.7, there exists an expanding cover of $\Lambda(\Gamma)$.
Furthermore, it is a controlled expanding cover due to the boundedness of
$g^{\prime}$ and $g^{\prime\prime}$. We finish the proof by Theorem 6.1. ∎

Theorem $1.1$ in [Me14] gives a local rigidity result about the limit sets of
Kleinian groups which are Schottky sets.

######  Theorem 6.10 (Merenkov).

Suppose that $\Gamma$ and $\tilde{\Gamma}$ are Kleinian groups whose limit
sets $S$ and $\tilde{S}$ are Schottky sets, respectively. We assume that
$\Gamma$ act on $S$ and $\tilde{\Gamma}$ act on $\tilde{S}$ as uniform
convergence groups. If $f:A\to S$ is a quasiconformal map defined on an open
(in relative topology) connected subset $A$ of $S$, then f has to be the
restriction of a Möbius transformation that takes $S$ onto $\tilde{S}$.

Recall that a _Schottky set_ is a compact subset of $\mathbb{S}^{2}$ whose
complement is a union of at least three open round discs whose closures have
empty intersection. The original theorem in [Me14] only requires that the
Kleinian groups act on their limit sets cocompactly on triples, but this is
equivalent to the group acting as a uniform convergence group. See [GM87] for
a reference.

Finally, we finish the proof of Theorem 1.4 by applying Thereon 1.3 and
Corollary 6.9 to Theorem 6.10.

###  6.3. Visual spheres of expanding Thurston maps

An expanding Thurston map $f:S^{2}\to S^{2}$ is a branched covering map on a
topological sphere with $\deg(f)\geq 2$ such that $f$ locally expands $S^{2}$.
We investigate these maps in this subsection and apply our results to visual
spheres of expanding Thurston maps. The main reference is [BM17].

Let $S^{2}$ be a topological sphere and $f:S^{2}\to S^{2}$ be a branched
covering map on $S^{2}$ with $\deg(f)\geq 2$. A point $p\in S^{2}$ is a
_critical point_ of $f$ if near $p$ the map $f$ is not a local homeomorphism.
We denote by $\textrm{crit}(f)$ the set of critical points of $f$ and $f^{n}$
the $n$-th iteration of $f$, then the set of _postcritical points_ of $f$ is
given by

$\mathrm{post}(f)=\bigcup_{n\geq 1}\\{f^{n}(p):p\in\textrm{crit}(f)\\}.$

We say $f$ is a _Thurston map_ if $f:S^{2}\to S^{2}$ is a branched covering
map with $\deg(f)\geq 2$ and $\mathrm{post}(f)$ is finite.

Let $f:S^{2}\to S^{2}$ be a Thurston map and $\mathcal{C}$ be a Jordan curve
in $S^{2}$ with $\mathrm{post}(f)\subset\mathcal{C}$. We fix a base metric $d$
on $S^{2}$ that induces the given topology on $S^{2}$. The Jordan curve
$\mathcal{C}$ divides $S^{2}$ into two sides and each of them(including the
boundary) is called a _$0$ -tile_. The preimages of $0$-tiles under $f^{-n}$
divide $S^{2}$ into what are called _$n$ -tiles_, and each $n$-tile is
contained in some $(n-1)$-tile. For $n\in\mathbb{N}$ we denote by
$\textrm{mesh}(f,n,\mathcal{C})$ the supremum of the diameters of all
$n$-tiles. A Thurston map $f:S^{2}\to S^{2}$ is _expanding_ if there exists a
Jordan curve $\mathcal{C}\subset S^{2}$ with
$\mathrm{post}(f)\subset\mathcal{C}$ such that
$\lim\limits_{n\to\infty}\textrm{mesh}(f,n,\mathcal{C})=0$. Notice that it is
a topological property, since expanding is independent of the choice of the
base metric.

For $x\neq y$, we define

$m(x,y)=\max\\{n\in\mathbb{N}\cup\\{0\\}:\exists\ \textrm{non-disjoint
$n$-tiles}\ X\textrm{and}\ Y\ \textrm{such that}\ x\in X,\ y\in Y\\}.$

If $x=y$, we define $m(x,x)=\infty$.

Let $f:S^{2}\to S^{2}$ be a Thurston map. A metric $\rho$ on $S^{2}$ is called
a _visual metric_ if there exists a Jordan curve $\mathcal{C}$ in $S^{2}$ with
$\mathrm{post}(f)\subset\mathcal{C}$, a parameter $\Lambda>1$ and
$C_{1},C_{2}>0$ such that

(22) $C_{1}\Lambda^{-m(x,y)}\leq\rho(x,y)\leq C_{2}\Lambda^{-m(x,y)}$

for all $x,y\in S^{2}$. Here we define $\Lambda^{-\infty}:=0$. The number
$\Lambda$ is called the _expansion factor_ of the visual metric and
$C_{1},C_{2}$ are independent of $x$ and $y$.

The following proposition shows some important properties of visual metrics on
$S^{2}$. Readers may refer to Chapter $8$, $16$ and $18$ of [BM17] for more
details.

######  Proposition 6.11.

Let $f:S^{2}\to S^{2}$ be an expanding Thurston map, then

  1. (1)

Every visual metric induces the given topology on $S^{2}$.

  2. (2)

There exists a $\Lambda_{0}>1$ such that for any $\Lambda\in(1,\Lambda_{0})$,
there exists a visual metric with expansion factor $\Lambda$ on $S^{2}$.

  3. (3)

Any two visual metrics are Hölder equivalent, and bi-Lipschitz equivalent if
they have the same expansion factor $\Lambda$.

  4. (4)

If $\rho$ is a visual metric for $f$, then $f:S^{2}\to S^{2}$ is a Lipschitz
map.

  5. (5)

Let $\rho$ be a visual metric for $f$. $(S^{2},\rho)$ is doubling if and only
if $f$ has no periodic critical points.

The visual sphere $S^{2}$ of an expanding Thurston map can be identified with
the boundary at infinity of a certain Gromov hyperbolic graph constructed from
tiles. Namely, let $f:S^{2}\to S^{2}$ be an expanding Thurston map and let
$\mathcal{C}\subset S^{2}$ be a Jordan curve with
$\mathrm{post}(f)\subset\mathcal{C}$, then we define an infinite graph
$G=G(f,\mathcal{C})$ using the associated tiles. The set of vertices of
$G(f,\mathcal{C})$ is given by the collection of tiles on all levels, where it
is convenient to add $S^{2}$ as a tile of level $-1$ and the base point of the
graph. One connects two vertices by an edge if the corresponding tiles have
non-empty intersection and their levels differ by at most $1$. We call it the
_tile graph_ and denote it by $G(f,\mathcal{C})$. The following proposition is
from Chapter $10$ of [BM17]:

######  Proposition 6.12.

Let $f:S^{2}\to S^{2}$ be an expanding Thurston map and $\mathcal{C}\subset
S^{2}$ be a Jordan curve with $\mathrm{post}(f)\subset\mathcal{C}$, then

  1. (1)

$G(f,\mathcal{C})$ is hyperbolic.

  2. (2)

$\partial_{\infty}G$ can naturally be identified with $S^{2}$. Under this
identification, a metric on $\partial_{\infty}G\cong S^{2}$ is visual in the
sense of Gromov hyperbolic spaces if and only if it is visual in the sense of
expanding Thurston maps.

  3. (3)

If $\mathcal{C}^{\prime}\subset S^{2}$ is another Jordan curve with
$\mathrm{post}(f)\subset\mathcal{C}^{\prime}$, then the graphs
$G(f,\mathcal{C})$ and $G(f,\mathcal{C}^{\prime})$ are quasi-isometric.

The following proposition focuses on a specific visual metric constructed in
Chapter $16$ of [BM17]. The original proof is mainly in Chapter $16$ of
[BM17]. Readers may also refer to [Wu19] for the details.

######  Proposition 6.13.

Let $f:S^{2}\to S^{2}$ be an expanding Thurston map, then there exists a
specific visual metric $\rho_{0}$ with expansion factor $\Lambda$ such that

  1. (1)

There exists a $C_{1}\in(0,1/\Lambda]$ such that if
$\rho_{0}(x,y)<C_{1}\cdot\min\\{\rho_{0}(f(x),p):p\in\mathrm{post}(f)\setminus\\{f(x)\\}\\}$,
then $\rho_{0}(f(x),f(y))=\Lambda(x,y)$.

  2. (2)

There exists a $C_{2}\in(0,C_{1}/\Lambda]$ such that for any $x\in
S^{2}\setminus\mathrm{post}(f)$, any $n\in\mathbb{N}$ and any
$\delta\leq\min\\{\rho_{0}(x,p):p\in\mathrm{post}(f)\\}$, if
$f^{n}(x^{\prime})=x$, then there exists a scaling map

$f^{n}:B(x^{\prime},\Lambda^{-n}C_{2}\delta)\to B(x,C_{2}\delta)$

with scaling factor $\Lambda^{n}$. More precisely, for any $y,z\in
B(x^{\prime},\Lambda^{-n}C_{2}\delta)$, we have
$\rho_{0}(f^{n}(y),f^{n}(z))=\Lambda^{n}\rho_{0}(y,z)$.

Notice that the specific visual metric $\rho_{0}$ defined in [Wu19] is
corresponding to an $f$-invariant Jordan curve $\mathcal{C}$ with
$\mathrm{post}(f)\subset\mathcal{C}$. However, the above proposition is still
valid for any Jordan curve that contains $\mathrm{post}(f)$ due to Theorem
$15.1$ in [BM17]. Readers may refer to Theorem $16.3$ in [BM17] for more
details.

We study the weak tangents at postcritical points in the following lemma.

######  Lemma 6.14.

Let $f:S^{2}\to S^{2}$ be an expanding Thurston map and $\rho$ be a visual
metric for $f$. Let $p\in\mathrm{post}(f)$, then for any weak tangent
$T_{p}S^{2}$ in $WT_{p}(S^{2})$, there exists a ball $B$ in $S^{2}$ such that
$B$ is quasisymmetrically embedded into $T_{p}S^{2}$.

###### Proof.

It is sufficient to proof Lemma 6.14 for $(S^{2},\rho_{0})$ where $\rho_{0}$
is the specific visual metric in Proposition 6.13.

Let $p$ be a postcritical point and let

$r_{0}=\frac{C_{1}}{2\Lambda}\min\\{\rho_{0}(p_{1},p_{2}):p_{1},p_{2}\in\mathrm{post}(f)\
\textrm{and}\ p_{1}\neq p_{2}\\}$

where $C_{2}$ is the constant in Proposition 6.13.

For any $x\in B(p,r_{0}),x\neq p$ and any $r<C_{2}\Lambda^{-2}\rho_{0}(x,p)$,
$f$ is a scaling map on $B(x,r)$ by Proposition 6.13. Furthermore, there
exists a $n\in\mathbb{N}$ such that $f^{n}$ expands $B(x,r)$ and maps $x$ into
$S^{2}\setminus\bigcup_{q\in\mathrm{post}(f)}B(q,r_{0})$.

For any $x^{\prime}\notin\bigcup_{q\in\mathrm{post}(f)}B(q,r_{0})$,
$f(x^{\prime})\notin\bigcup_{q\in\mathrm{post}(f)}B(q,r_{0})$ due to the
expansion of $f$ near $\mathrm{post}(f)$. Thus for any
$r<C_{2}\Lambda^{-1}r_{0}$, $f$ is a scaling map on $B(x^{\prime},r)$ with
scaling factor $\Lambda$ by Proposition 6.13. This shows that $S^{2}$ is
quasi-self-symmetric at any $x\in B(p,r_{0})$ with the standard size
$C_{2}\Lambda^{-2}\rho_{0}(x,p)$ and the bound $C_{2}\Lambda^{-1}r_{0}$.

Applying Theorem 5.5 to $(S^{2},\rho_{0})$ finishes the proof.

∎

We are able to prove Theorem 1.5 now. Notice that Theorem 1.5 has been proved
in [Wu19] with ideas from dynamics, but we can give an alternative proof based
on tools generated in this paper. The theorem proved in [Wu19] is not exactly
the same as Theorem 1.5, since the definition of weak tangents between our
paper and [Wu19] is different. The proof is rather intuitive except the
implication $(3)\Longrightarrow(4)$. We included a complete proof of this
theorem for the convenience of the readers.

###### Proof of Theorem 1.5.

$(1)\Longrightarrow(2)$: $S^{2}$ is doubling due to Proposition 6.11; thus
finishes the proof by Theorem 1.1.

$(2)\Longrightarrow(3)$: It is trivial.

$(3)\Longrightarrow(4)$: Let $T_{p}X$ be the weak tangent that is
quasisymmetric to $\mathbb{R}^{2}$.

If $p\notin\mathrm{post}(f)$, then $S^{2}$ is quasi-self-symmetric at a
sufficiently small neighborhood of $p$ due to Proposition 6.13(more precisely,
the proof of Lemma 6.14). Furthermore, the standard sizes and the bounds of
these points are uniformly bounded. We finish the proof by Corollary 5.3.

If $p\in\mathrm{post}(f)$, it is proved by Lemma 6.14.

$(4)\Longrightarrow(1)$: Let $\mathcal{C}$ be the Jordan curve in $S^{2}$
where the visual metric $\rho$ is constructed corresponding to $f$ and
$\mathcal{C}$. Let $U_{1},U_{2}$ be the two $0$-tiles of $S^{2}$. Since $f$ is
expanding, there exists two simply-connected closed subsets
$U^{\prime}_{1},U^{\prime}_{2}\in U$ and $n\in\mathbb{N}$ such that
$f^{n}:U^{\prime}_{1}\to U_{1}$ and $f^{n}:U^{\prime}_{2}\to U_{2}$ are
homeomorphisms. Furthermore, they are also bi-Lipschitz by the definition of
visual metrics. Thus there exist two quasisymmetric embeddings
$f_{1}:U_{1}\to\mathbb{R}^{2}$ and $f_{2}:U_{2}\to\mathbb{R}^{2}$. Notice that
$\mathcal{C}$ is a quasi-circle by Theorem $15.1$ and $15.3$ in [BM17]. Thus,
by post-composing quasisymmetries, we may assume that
$f_{1}:U_{1}\to\overline{\mathbb{D}}$ and
$f_{2}:U_{2}\to\overline{\mathbb{D}}$ are two quasisymmetric maps. Here
$\overline{\mathbb{D}}$ is the closed unit disk. Since $f_{1}\circ f_{2}^{-1}$
is quasisymmetric on $\partial\mathbb{D}$, post-composing $f_{2}$ by a
quasisymmetry again, we may assume that
$f_{1}|_{\mathcal{C}}=f_{2}|_{\mathcal{C}}$. The quasisymmetries being post-
composed above are constructed by Beurling-Ahlfors extension. Readers may
refer to Section $5$ of [Bo11] for a reference. Notice that
$\overline{\mathbb{D}}$ is bi-Lipschitz to the closed half sphere. By gluing
$f_{1}$ and $f_{2}$, we obtain a homeomorphism $f:S^{2}\to\mathbb{S}^{2}$.
What is left to prove is that $f$ is a quasisymmetry.

Assume that $f_{1},f_{2}$ are $\eta$-quasisymmetric. Let $x,y,z$ be distinct
points in $S^{2}$, then it is sufficient to prove that

$\frac{|f(x)-f(y)|}{|f(x)-f(z)|}\leq\eta_{1}\left(\frac{\rho_{0}(x,y)}{\rho_{0}(x,z)}\right)$

where $\rho_{0}$ is the specific visual metric in Proposition 6.13 and
$\eta_{1}$ depends only on $\eta$. We split the proof into three situations.

Let $x,z\in U_{1}$ and $y\in U_{2}$. By the definition of $\rho_{0}$, there
exists a point $w\in\mathcal{C}$ such that
$\rho_{0}(x,y)=\rho_{0}(x,w)+\rho_{0}(w,y)$.

If $|f(x)-f(w)|\geq\frac{1}{3}|f(x)-f(y)|$, then

$\frac{|f(x)-f(y)|}{|f(x)-f(z)|}\leq\frac{3|f(x)-f(w)|}{|f(x)-f(z)|}\leq
3\eta\left(\frac{\rho_{0}(x,w)}{\rho_{0}(x,z)}\right)\leq
3\eta\left(\frac{\rho_{0}(x,y)}{\rho_{0}(x,z)}\right).$

If $|f(x)-f(w)|<\frac{1}{3}|f(x)-f(y)|$, we choose a point $v$ on
$\mathcal{C}$ such that $|f(v)-f(w)|=|f(y)-f(w)|$. Notice that such a point
exists since $f(\mathcal{C})$ is a unit circle. Then

$\frac{|f(x)-f(y)|}{|f(x)-f(z)|}\leq\frac{3|f(y)-f(w)|}{2|f(x)-f(z)|}=\frac{3|f(v)-f(w)|}{2|f(x)-f(z)|}$

Since
$|f(x)-f(v)|\geq|f(v)-f(w)|-|f(x)-f(w)|\geq|f(v)-f(w)|-1/2|f(v)-f(w)|=1/2|f(v)-f(w)|$,
we have

$\frac{|f(x)-f(y)|}{|f(x)-f(z)|}\leq\frac{3|f(x)-f(v)|}{|f(x)-f(z)|}\leq
3\eta\left(\frac{\rho_{0}(x,v)}{\rho_{0}(x,z)}\right)\leq
3\eta\left(\frac{\rho_{0}(x,w)+\rho_{0}(w,v)}{\rho_{0}(x,z)}\right).$

Since

$\frac{|f(y)-f(w)|}{|f(v)-f(w)|}\leq\eta\left(\frac{\rho_{0}(y,w)}{\rho_{0}(v,w)}\right),$

$\rho_{0}(v,w)\leq 1/\eta^{-1}(1)\cdot\rho_{0}(y,w)$. We denote by
$\lambda:=\max\\{1,1/\eta^{-1}(1)\\}$.

Thus

$\displaystyle\frac{|f(x)-f(y)|}{|f(x)-f(z)|}$ $\displaystyle\leq$
$\displaystyle
3\eta\left(\frac{\rho_{0}(x,w)+\rho_{0}(w,v)}{\rho_{0}(x,z)}\right)$
$\displaystyle\leq$ $\displaystyle
3\eta\left(\lambda\frac{\rho_{0}(x,w)+\rho_{0}(y,w)}{\rho_{0}(x,z)}\right)$
$\displaystyle=$ $\displaystyle
3\eta\left(\lambda\frac{\rho_{0}(x,y)}{\rho_{0}(x,z)}\right).$

Let $x,y\in U_{1}$ and $z\in U_{2}$. Similarly, there exists a point
$w^{\prime}\in\mathcal{C}$ such that
$|f(x)-f(z)|=|f(x)-f(w^{\prime})|+|f(w^{\prime})-f(z)|$. We also split the
proof into two cases: $\rho_{0}(x,w^{\prime})\geq C\cdot\rho_{0}(x,z)$ or not,
where $C$ is a sufficiently small constant that depends on $\mathcal{C}$. The
proof of this situation is literally the same as above.

If $\rho_{0}(x,w^{\prime})\geq C\cdot\rho_{0}(x,z)$, then

$\frac{\rho_{0}(x,y)}{\rho_{0}(x,z)}\geq
C\frac{\rho_{0}(x,y)}{\rho_{0}(x,w^{\prime})}\geq
C\eta^{-1}\left(\frac{|f(x)-f(y)|}{|f(x)-f(w^{\prime})|}\right)\geq
C\eta^{-1}\left(\frac{|f(x)-f(y)|}{|f(x)-f(z)|}\right).$

If $\rho_{0}(x,w^{\prime})<C\cdot\rho_{0}(x,z)$, we choose a point
$v^{\prime}$ on $\mathcal{C}$ such that
$\rho_{0}(v^{\prime},w^{\prime})=3C\cdot\rho_{0}(z,w^{\prime})$. Notice that
such a point exists since $\mathcal{C}$ is a quasi-circle and $C$ is
sufficiently small. Then

$\frac{\rho_{0}(x,y)}{\rho_{0}(x,z)}\geq(1-C)\cdot\frac{\rho_{0}(x,y)}{\rho_{0}(z,w^{\prime})}=(3C-3C^{2})\cdot\frac{\rho_{0}(x,y)}{\rho_{0}(v^{\prime},w^{\prime})}.$

Since $C$ is sufficiently small, we assume that
$\rho_{0}(x,v^{\prime})\geq\rho_{0}(v^{\prime},w^{\prime})-\rho_{0}(x,w^{\prime})\geq\rho_{0}(v^{\prime},w^{\prime})-1/(3-3C)\cdot\rho_{0}(v^{\prime},w^{\prime})\geq
1/2\rho_{0}(v^{\prime},w^{\prime})$. Then

$\displaystyle\frac{\rho_{0}(x,y)}{\rho_{0}(x,z)}$ $\displaystyle\geq$
$\displaystyle\frac{3C-3C^{2}}{2}\frac{\rho_{0}(x,y)}{\rho_{0}(x,v^{\prime})}$
$\displaystyle\geq$
$\displaystyle\frac{3C-3C^{2}}{2}\eta^{-1}\left(\frac{|f(x)-f(y)|}{|f(x)-f(v^{\prime})|}\right)$
$\displaystyle\geq$
$\displaystyle\frac{3C-3C^{2}}{2}\eta^{-1}\left(\frac{|f(x)-f(y)|}{|f(x)-f(w^{\prime})|+|f(w^{\prime})-f(v^{\prime})|}\right).$

Since

$\frac{|f(v^{\prime})-f(w^{\prime})|}{|f(z)-f(w^{\prime})|}\leq\eta\left(\frac{\rho_{0}(v,w^{\prime})}{\rho_{0}(z,w^{\prime})}\right),$

$|f(v^{\prime})-f(w^{\prime})|\leq\eta(3C)\cdot|f(z)-f(w^{\prime})|$. We
denote by $\lambda^{\prime}:=\max\\{1,\eta(3C)\\}$.

Thus

$\displaystyle\frac{\rho_{0}(x,y)}{\rho_{0}(x,z)}$ $\displaystyle\geq$
$\displaystyle\frac{3C-3C^{2}}{2}\eta^{-1}\left(\frac{|f(x)-f(y)|}{|f(x)-f(w^{\prime})|+|f(w^{\prime})-f(v^{\prime})|}\right)$
$\displaystyle\geq$
$\displaystyle\frac{3C-3C^{2}}{2}\eta^{-1}\left(\frac{1}{\lambda^{\prime}}\frac{|f(x)-f(y)|}{|f(x)-f(w^{\prime})|+|f(z)-f(w^{\prime})|}\right)$
$\displaystyle=$
$\displaystyle\frac{3C-3C^{2}}{2}\eta^{-1}\left(\frac{1}{\lambda^{\prime}}\frac{|f(x)-f(y)|}{|f(x)-f(z)|}\right).$

Let $x\in U_{1}$ and $y,z\in U_{2}$. The proof of this situation relies on the
above results and it also follows the same idea. We scratch the proof below
without details. Similarly, let $w\in\mathcal{C}$ where
$\rho_{0}(x,y)=\rho_{0}(x,w)+\rho_{0}(w,y)$ and $v\in\mathcal{C}$ where
$|f(v)-f(w)|=|f(y)-f(w)|$.

If $|f(x)-f(w)|\geq\frac{1}{3}|f(x)-f(y)|$, then

$\frac{|f(x)-f(y)|}{|f(x)-f(z)|}\leq\frac{3|f(x)-f(w)|}{|f(x)-f(z)|}\leq
3\eta_{0}\left(\frac{\rho_{0}(x,w)}{\rho_{0}(x,z)}\right)\leq
3\eta_{0}\left(\frac{\rho_{0}(x,y)}{\rho_{0}(x,z)}\right).$

where $\eta_{0}$ depends only on $\eta$.

If $|f(x)-f(w)|<\frac{1}{3}|f(x)-f(y)|$, then

$\frac{|f(x)-f(y)|}{|f(x)-f(z)|}\leq\frac{3|f(x)-f(v)|}{|f(x)-f(z)|}\leq
3\eta_{0}\left(\frac{\rho_{0}(x,v)}{\rho_{0}(x,z)}\right)\leq
3\eta_{0}\left(\frac{\rho_{0}(x,w)+\rho_{0}(w,v)}{\rho_{0}(x,z)}\right).$

Thus

$\frac{|f(x)-f(y)|}{|f(x)-f(z)|}\leq
3\eta_{0}\left(\max\\{1,1/\eta^{-1}(1)\\}\cdot\frac{\rho_{0}(x,y)}{\rho_{0}(x,z)}\right).$

∎

## Acknowledgments

The author wants to thank Sergiy Merenkov who inspired this problem and gave
helpful advice. Jeremy Tyson talked with the author and suggested the
uniformly perfectness in proving Theorem 1.1. Ilya Kapovich talked with the
author about hyperbolic groups and the idea which is used in Subsection 6.2.

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