††thanks: These authors contribute equally to this work.††thanks: These
authors contribute equally to this work.

# Non-chiral non-Bloch invariants and topological phase diagram in non-unitary
quantum dynamics without chiral symmetry

Yue Zhang  Ministry of Education Key Laboratory for Nonequilibrium Synthesis
and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  Shuai Li  Ministry of Education Key
Laboratory for Nonequilibrium Synthesis and Modulation of Condensed
Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an
710049, China  Yingchao Xu  Ministry of Education Key Laboratory for
Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province
Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,
School of Physics, Xi’an Jiaotong University, Xi’an 710049, China  Rui Tian
Ministry of Education Key Laboratory for Nonequilibrium Synthesis and
Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  Miao Zhang  Ministry of Education
Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed
Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum
Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an
710049, China  Hongrong Li  Ministry of Education Key Laboratory for
Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province
Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,
School of Physics, Xi’an Jiaotong University, Xi’an 710049, China  Hong Gao
Ministry of Education Key Laboratory for Nonequilibrium Synthesis and
Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  M. Suhail Zubairy  Institute for
Quantum Science and Engineering (IQSE) and Department of Physics and
Astronomy, Texas A&M University, College Station, TX 77843-4242, USA  Fuli Li
Ministry of Education Key Laboratory for Nonequilibrium Synthesis and
Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  Bo Liu<EMAIL_ADDRESS>Ministry
of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of
Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and
Quantum Optoelectronic Devices, School of Physics, Xi’an Jiaotong University,
Xi’an 710049, China

###### Abstract

The non-Bloch topology leads to the emergence of various counter-intuitive
phenomena in non-Hermitian systems under the open boundary condition (OBC),
which can not find a counterpart in Hermitian systems. However, in the non-
Hermitian system without chiral symmetry, being ubiquitous in nature,
exploring its non-Bloch topology has so far eluded experimental effort. Here
by introducing the concept of non-chiral non-Bloch invariants, we
theoretically predict and experimentally identify the non-Bloch topological
phase diagram of a one-dimensional (1D) non-Hermitian system without chiral
symmetry in discrete-time non-unitary quantum walks of single photons.
Interestingly, we find that such topological invariants not only can
distinguish topologically distinct gapped phases, but also faithfully capture
the corresponding gap closing in open-boundary spectrum at the phase boundary.
Different topological regions are experimentally identified by measuring the
featured discontinuities of the higher moments of the walker’s displacement,
which amazingly match excellently with our defined non-Bloch invariants. Our
work provides a useful platform to study the interplay among topology,
symmetries and the non-Hermiticity.

Non-Hermiticity is ubiquitous in nonconservative systems, such as in open
quantum systems Rotter (2009), correlated electron systems Shen and Fu (2018),
and systems with gain or loss Shen and Fu (2018); Konotop et al. (2016);
Regensburger et al. (2012); Zhen et al. (2015); Chen et al. (2017); Zhou et
al. (2018); Tang et al. (2020); Zhang et al. (2020); Ji et al. (2020).
Distinct from conventional Hermitian Hamiltonians, non-Hermitian matrices
exhibit unconventional characteristics, where eigenstates in general are
nonorthogonal and a complex-energy spectrum exists Moiseyev (2011).
Unprecedented phenomena with no counterparts in Hermitian systems have been
unveiled, including unidirectional invisibility Feng et al. (2013),
exceptional-point encirclement Gao et al. (2015); Doppler et al. (2016); Xu et
al. (2016) and enhanced sensitivity Chen et al. (2017); Hodaei et al. (2017);
Wiersig (2014). In particular, recently the interplay between non-Hermiticity
and topology has attracted tremendous attention. Their union has resulted in a
wide variety of intriguing non-Hermitian phenomena, being discovered and
reinterpreted within the framework of topology Bergholtz et al. (2021);
Kawabata et al. (2019); Ashida et al. (2020). One of the particularly
captivating examples is the breakdown of conventional bulk-boundary
correspondence in non-Hermitian systems under OBC Hasan and Kane (2010); Qi
and Zhang (2011). Recent researches establish a non-Bloch band theory. Based
on a generalized Brillouin zone (GBZ), non-Bloch topological invariants are
introduced and the bulk-boundary correspondence can be restored Yao and Wang
(2018); Yao et al. (2018); Yokomizo and Murakami (2019); Yang et al. (2020).
Such quantized topological invariants are fundamentally linked to the symmetry
of the system. Specifically, much attention has been paid to the systems with
chiral (or equivalent) symmetry Xiao et al. (2020); Mochizuki et al. (2020,
2016); Xiao et al. (2017). However, the non-Bloch topology in non-Hermitian
systems without chiral symmetry has never been experimentally demonstrated yet
in any system.

Figure 1: Non-Bloch topological phase diagram of our 1D non-chiral non-
Hermitian system. (a) Different topological phases characterized by our
defined non-chiral non-Bloch topological invariant $v$ as the function of the
coin parameter $({\theta_{1}}-{\theta_{2}})$ with a fixed ${e^{\gamma}}=0.82$.
The gapped topological trivial and non-trivial regions are characterized with
$v=0$ and $v=\pm 1$, respectively. The exceptional regions are characterized
with $v$ being the half integer. (b) The bulk spectrum $E$ of our non-chiral
non-unitary quantum walk in a 1D lattice under different generalized boundary
conditions (GBCs) through choosing various $\Delta$. (c) The generalized
Brillouin zone (GBZ) for our non-chiral non-unitary quantum walk under
different GBCs. For each nonvanished $\Delta$, the GBZ under GBC consists of
two circles. One is outside the GBZ when $\Delta=0$ (red circle) and the other
is inside that. In (b) and (c), we choose $\theta_{1}=0.6\pi$ and
$\theta_{2}=0.58\pi$. Other parameters are same as in (a).

Here, we theoretically predict and experimentally identify the non-Bloch
topology of a 1D non-Hermitian system without chiral symmetry in discrete-time
non-unitary quantum walks of single photons. Such a study is motivated by the
recent rapid progresses in exploring non-Hermitian topological phases of
matter through quantum walks in various synthetic systems, including photonics
Xiao et al. (2017); Zhan et al. (2017); Wang et al. (2019, 2021); Xiao et al.
(2021); Chen et al. (2022, 2023) and cold atoms Xie et al. (2020). Distinct
from previous studies of 1D non-Hermitian system with chiral (or equivalent)
symmetry Xiao et al. (2020); Mochizuki et al. (2020, 2016); Xiao et al.
(2017), here we explore the non-Bloch topology of non-Hermitian system without
chiral symmetry. First, we introduce the concept of non-chiral non-Bloch
invariants for our non-unitary quantum walks. Although the chiral symmetry is
absent, quantized topological invariants can still be defined in the complex
momentum plane taking into account of the deviations of localized bulk states
from Bloch waves. Interestingly, we find that such topological invariants not
only can characterize topologically distinct gapped phases, but also
faithfully capture the corresponding gap closing in open-boundary spectrum at
the phase boundary. Therefore, the non-Bolch topological phase diagram can be
unambiguously identified by them. Second, we experimentally determine the
topological phase transition of our 1D non-chiral non-Hermitian system through
measuring the featured discontinuities of the higher moments of the walker’s
displacement Cardano et al. (2016, 2017), which amazingly match excellently
with our defined non-Bloch invariants. Therefore, the non-Bolch topological
phase diagram of such a non-chiral non-Hermitian system can be identified.

Non-chiral non-unitary quantum walk — We study a non-chiral non-unitary
quantum walk in a 1D lattice, which can be captured by the following Floquet
operator

$U_{0}=T_{\downarrow}R_{y}(\theta_{2})MT_{\uparrow}R_{y}(\theta_{1}),$ (1)

where ${R_{y}}(\theta)$ stands for the coin operator, referring to rotate coin
states by $\theta$ about the y-axis. ${T_{\downarrow}}({T_{\uparrow}})$ labels
the pseudospin-dependent translation by one lattice site, where $\downarrow$
and $\uparrow$ represent the coin state $\left|0\right\rangle$ and
$\left|1\right\rangle$ , respectively.
$M=\mathbb{I}_{x}\otimes(e^{\gamma}\left|\downarrow\right\rangle\left\langle\downarrow\right|+e^{-\gamma}\left|\uparrow\right\rangle\left\langle\uparrow\right|)$
is the polarization selective-loss operator with $\gamma$ being a tunable
parameter in experiments and $\mathbb{I}_{x}$ being the identity matrix in
lattice modes, which introduces the non-unitarity.

Figure 2: Experimental implementation. A laser field is attenuated to the
single-photon level via neutral density filters and coupled into a
polarization dependent optical loop through a beam splitter (BS) with a
splitting ratio of $5/95$. The coin operator is implemented by the half wave
plate (HWP) and polarization dependent optical delay was realized by two PBS
(polarizing beam splitter) loops, where the free-space path difference between
horizontally and vertically polarized photons is 60 cm. M operator is
implemented by adding an attenuator with attenuation rate ${e^{\gamma}}=0.82$
to the path of horizontally polarized photons. After using a prism to reflect
the pulse back into the loop, at last around $5\%$ of photons are reflected by
a beam splitter for detection and the transmitted photons continue to
propagate through the interferometric network. Figure 3: The probability
distribution for the walker. (a) and (c) Time-dependent position probability
distribution of a seven-step non-unitary quantum walk. (b) and (d) are the
corresponding distribution at the last step. Here the walker is initially
prepared at $x=0$ with the coin state $\left|1\right\rangle$. In (a) and (b),
$\theta_{1}=0.11\pi$, while in (c) and (d), $\theta_{1}=0.56\pi$. Other
parameters are same as in Fig. 1.

To construct the non-chiral non-Bloch band theory describing the non-unitary
quantum walk governed by ${U_{0}}$ , we rewrite the eigenstate of ${U_{0}}$ as

$\left|\psi\right\rangle=\sum\limits_{j,n=1,2}\beta_{n}^{j}\left|j\right\rangle\otimes\left|\phi_{n}\right\rangle,$
(2)

where ${\beta_{n}}$ is the spatial-mode function for the $n$-th mode, which
takes into account of the generic deviations of localized bulk states from
Bloch waves. $\left|{{\phi_{n}}}\right\rangle$ is the corresponding coin state
and $j$ stands for the lattice site. For the non-unitary quantum walk
considered here, it is shown that there are two spatial modes labeled by
$n=1,2$ for each bulk state (see details in Supplementary Material(SM)). A
Floquet system captured by the time-evolution operator ${U_{0}}$ corresponds
to an effective non-Hermitian Hamiltonian ${H_{eff}}$ via the relation
${U_{0}}={e^{-i{H_{eff}}}}$ , where quantum walk dynamics under ${U_{0}}$ can
be regarded as a stroboscopic simulation of the non-unitary dynamics driven by
${H_{eff}}$. Following the non-Bloch band theory under OBC, we can express
${H_{eff}}$ on the GBZ. This can be achieved by replacing the Bloch phase
factor ${e^{ik}}$ with $\beta$ on the GBZ, which manifests itself as a closed
loop on the complex plane. $\beta$ can thus be regarded as the generalized
quasi-momentum in the non-Bloch band theory Yao and Wang (2018); Yao et al.
(2018); Yokomizo and Murakami (2019); Yang et al. (2020). Then, ${H_{eff}}$
can be expressed as $\mathbf{\tilde{h}}(\beta)\cdot\mathbf{\sigma}$ with
$\mathbf{\sigma}$ being the Pauli vector (see details in SM) and

$\displaystyle\tilde{h}_{x}$ $\displaystyle=-\frac{E}{2i\sin E}(\beta
a-\beta^{-1}a^{-1})\cos\theta_{2}/2$ $\displaystyle\tilde{h}_{y}$
$\displaystyle=-\frac{E}{2i\sin
E}[i(a^{-1}+a)\cos\theta_{1}/2\sin\theta_{2}/2$ $\displaystyle+i(\beta
a+\beta^{-1}a^{-1})\sin\theta_{1}/2\cos\theta_{2}/2]$
$\displaystyle\tilde{h}_{z}$ $\displaystyle=\frac{E}{2i\sin
E}(a-a^{-1})\sin\theta_{2}/2$ (3)

with $E\equiv
E_{\pm}=\pm\arccos(-\frac{\alpha+\alpha^{-1}}{2}\sin\theta_{1}/2\sin\theta_{2}/2+\frac{\beta\alpha+\beta^{-1}\alpha^{-1}}{2}\cos\theta_{1}/2\cos\theta_{2}/2)$
and $a=e^{\gamma}$. Distinct from previous studies Xiao et al. (2020);
Mochizuki et al. (2020, 2016); Xiao et al. (2017), all the three components,
i.e., ${\tilde{h}_{x(y,z)}}$ of ${\bf\tilde{h}}(\beta)$, are nonvanishing and
the chiral symmetry is absent here.

Non-chiral non-Bloch topological invariants — To find out the topological
invariant, that not only can pick out topologically distinct gapped phases,
but also can determine the phase boundary through the corresponding gap
closing in open-boundary spectrum for our non-unitary quantum walks, we
consider adding the generalized boundary condition (GBC) Guo et al. (2021),
which can be captured by the following pseudospin-dependent translation

$T_{\uparrow}^{\prime}=T_{\uparrow}+\Delta\left|1\right\rangle\left\langle
L\right|\otimes\left|\uparrow\right\rangle\left\langle\uparrow\right|$
$T_{\downarrow}^{\prime}=T_{\downarrow}+\Delta\left|L\right\rangle\left\langle
1\right|\otimes\left|\downarrow\right\rangle\left\langle\downarrow\right|$ (4)

where $\Delta$ is a real number, describing the position shift at the
boundary. Then, our non-chiral non-unitary quantum walk in a 1D lattice under
GBC can be captured by the following Floquet operator

$U=T_{\downarrow}^{\prime}R_{y}(\theta_{2})MT_{\uparrow}^{\prime}R_{y}(\theta_{1}).$
(5)

It is worthy to note that when $\Delta=0,U={U_{0}}$ describes our studied non-
chiral non-unitary quantum walk under OBC. While $\Delta=1$, $U$ describes the
system under the periodic boundary condition (PBC). Therefore, when tuning
$\Delta$ in $U$, we can continuously change the boundary condition from PBC to
OBC. As shown in Fig. 1(b), the continuum spectrum of the effective
Hamiltonian determined by $U$ can be obtained through diagonalising $U$ in the
real space. Then, we can obtain the spatial-mode function ${\beta_{n}}$
associated with the corresponding continuum spectrum $E$ through the following
relation (see details in SM)

$\beta_{n}^{2}-\frac{2(\cos
E+\frac{\alpha+\alpha^{-1}}{2}\sin\theta_{1}/2\sin\theta_{2}/2)}{\alpha\cos\theta_{1}/2\cos\theta_{2}/2}\beta_{n}+\alpha^{-2}=0$
(6)

Using the obtained ${\beta_{n}}$, the GBZ under GBC can be constructed. As
shown in Fig. 1(c), for each nonvanished $\Delta$ , the spatial-mode function
${\beta_{n}}$ associated with the continuum spectrum forms two circles,
construing the corresponding GBZ under GBC. When $\Delta\to 0$, these GBZ will
form two paths towards the GBZ (red circle in Fig. 1(c)) under OBC: (i) the
path $C^{\text{outside}}_{\beta}$ outside the circle of GBZ under OBC, (ii)
the path $C^{\text{inside}}_{\beta}$ inside the circle of GBZ under OBC. Then,
the non-chiral non-Bloch topological invariant to describe our non-unitary
quantum walk captured by ${U_{0}}$ under OBC can be defined as

$v=\frac{1}{2\pi}\sum\nolimits_{\pm}\oint_{C^{\text{inside}}_{\beta}(\Delta\rightarrow
0)}d\beta\left\langle\psi_{\pm}^{L}\right|i\partial_{\beta}\left|\psi_{\pm}^{R}\right\rangle$
(7)

where $\left|\psi_{\pm}^{R}\right\rangle=\frac{[\sin E_{\pm}(E_{\pm}-\sin
2E_{\pm}h_{y})+\sin E_{\pm}^{\ast}(E_{\pm}+\sin
2E_{\pm}h_{y})]^{1/2}}{E_{\pm}\sqrt{2(E_{\pm}-h_{z})}}$
$(h_{x}-ih_{y},E_{\pm}-h_{z})^{T}$ and
$\left\langle\psi_{\pm}^{L}\right|=\frac{[\sin E_{\pm}(E_{\pm}-\sin
2E_{\pm}h_{y})+\sin E_{\pm}^{\ast}(E_{\pm}+\sin
2E_{\pm}h_{y})]^{-1/2}}{\sqrt{2(E_{\pm}-h_{z})}}(h_{x}+ih_{y}$
$,E_{\pm}-h_{z})$ are the eigenstates of the effective non-Hermitian
Hamiltonian determined by $U$. Interestingly, we find that the above defined
quantized topological invariant not only can characterize topologically
distinct gapped phases, but also faithfully capture the corresponding gap
closing in open-boundary spectrum at the phase boundary for our non-chiral
non-unitary quantum walk. As shown in Fig. 1(a), we show the non-Bloch
topological phase diagram on the ${\theta_{1}}-{\theta_{2}}$ plane
characterized with our defined topological invariant $v$. It is shown that the
gapped topological trivial and non-trivial regions are characterized with
$v=0$ and $v=\pm 1$, respectively. The gap closing in open-boundary spectrum
at the phase boundary can also be faithfully captured by the transition of $v$
from integer to half integer. The exceptional regions are thus characterized
with $v$ being the half integer. It is worth noting that one can also choose
the outside path $C^{\text{outside}}_{\beta}$ to define the topological
invariant. The same results will be obtained, i.e., the topological trivial,
non-trivial and exceptional regions are characterized with $v$ being 0, $\pm
1$ and half integer, respectively.

Figure 4: The probability distribution moments reveal non-chiral non-Bloch
topological phase transitions. (a) Measured probability distribution moments
(red dots) $M_{0.1}/N^{0.1}$ after a seven-step quantum walk as a function of
${\theta_{1}}$. Solid line represents the corresponding results of numerical
simulations. The dashed line is determined by the asymptotical result as
$N\to\infty$. The emergence of the abrupt slope variation appears at each
transition point between different regions characterized with distinct non-
chiral non-Bloch topological invariants $v$, which can thus be considered as
an experimental visible signature of the underlying topological phase
transition. (b) Numerical simulations of $M_{0.1}/N^{0.1}$ as a function of
${\theta_{1}}$. It is shown that, for $M_{0.1}$, even considering small steps
quantum-walk, for instance, $N=7$, the slope discontinuity of that is
consistent with the asymptotic limit $N\to\infty$. (c) Numerical simulations
of $M_{l}/N^{l}$ as a function of ${\theta_{1}}$. Here we choose $N^{l}$ as
the normalization factor and $N=80$. It is shown that different probability
distribution moments $M_{l}$ with distinct $l$ show the same slope
discontinuity in the asymptotic limit, which thus reveal the same topological
phase transition points. Other parameters are same as in Fig. 3.

Experimental non-Bloch topological phase diagram — We experimentally
investigate the non-unitary quantum-walk dynamics governed by ${U_{0}}$
through implementation of a photonic discrete-time quantum walk (DTQW) via
temporal-mode multiplexing scheme Schreiber et al. (2011). The experimental
setup is illustrated in Fig. 2. The position of the walker (photon) is encoded
into the arrival time at the detector. It is coupled to the coin state
realized by the horizontally and vertically polarized photons, respectively,
which are labeled with $\left|0\right\rangle$ and $\left|1\right\rangle$. The
$M$ operator is implemented by adding an attenuator to the path of
horizontally polarized photons (see details in SI). In our experimental setup
${e^{\gamma}}$ in $M$ is $0.82$ .

To capture the topological quantum transition for our non-chiral non-unitary
quantum walk, we employ the method of monitoring the probability distribution
moments of the walker position, where a slope discontinuity demonstrates the
transition point [39-40]. The moments of the probability distribution $P(m)$
associated with the walker position $m$ are defined as
$M_{l}=\sum\limits_{m}|m|^{l}P(m)$ . As shown in Fig. 4(c), the numerical
simulations show that in the infinite-steps-limit, when varying $\theta_{1}$,
at each phase transition point between different regions (in Fig. 1(a))
characterized with distinct $v$, ${M_{l}}$ undergoes a abrupt slope variation.
Therefore, the phase boundary determined by distinct $v$ is consistent with
the slope discontinuity of ${M_{l}}$. For a finite number of steps, we find
that ${M_{l}}$ have a continuous behavior converging to the asymptotical
result as the step $N\to\infty$. In particular, as shown in Fig. 4(b), for
$M_{0.1}$, this convergence is visible for values of $N$ that are small enough
to be achieved in our non-chiral non-unitary quantum-walk.

In experiments, we study seven-step quantum-walk dynamics and focus on
experimentally determining the topological phase transition of our non-chiral
non-unitary quantum-walk. Initializing the walker at $x=0$ with the coin state
$\left|1\right\rangle$, we measure the position probability distributions of a
seven-step quantum walk, both the time-dependent probability distribution and
the distribution at the last step are shown in Fig. 3. Based on that, we can
experimentally determine ${M_{l}}$. As shown in Fig. 4(a), experimental data
matched well with the numerical simulations. As expected in numerics (Fig.
4(b)), even considering seven-step quantum-walk, i.e., $N=7$, the slope
discontinuity of $M_{0.1}$ is consistent with the asymptotic limit
$N\to\infty$. Therefore, in our experimental detection as shown in Fig. 4(a),
when varying ${\theta_{1}}$ with a certain ${\theta_{2}}$, the emergence of
the abrupt slope variation in $M_{0.1}$ appears at each transition point
between different regions (marked by different colors) characterized with
distinct non-chiral non-Bloch topological invariants $v$, which is amazingly
consistent with the phase boundary determined by $v$. The observed non-
analyticity in $M_{0.1}$ can thus be considered as an experimental visible
signature of the underlying topological quantum transitions. Therefore, when
further varying ${\theta_{2}}$, the phase diagram on the
${\theta_{1}}-{\theta_{2}}$ plane as shown in Fig. 1(a) can be experimentally
identified.

Discussion & Conclusion — We have introduced the concept of non-chiral non-
Bloch invariants. Through using such topological invariants, we have
experimentally identified the non-Bloch topological phase diagram of a 1D non-
Hermitian system without chiral symmetry in discrete-time non-unitary quantum
walks of single photons. Even the chiral symmetry is absent, the quantized
non-chiral non-Bloch invariants can still be defined. It not only can
characterize topologically distinct gapped phases, but also faithfully capture
the corresponding gap closing in open-boundary spectrum at the phase boundary.
Through investigating the featured discontinuities of the higher moments of
the walker’s displacement, a non-chiral non-Bloch topological phase diagram
for our non-unitary quantum walks is experimentally identified, amazingly
matched with our defined non-Bloch invariants excellently. In prospect, our
approach should be valuable for advancing the understanding of topological
phenomena in open systems.

Acknowledgment — This work is supported by the National Key R$\&$D Program of
China (2021YFA1401700), NSFC (Grants No. 12074305, 12147137, 11774282), the
Fundamental Research Funds for the Central Universities (Grant No.
xtr052023002) and the Shaanxi Fundamental Science Research Project for
Mathematics and Physics (Grant No. 23JSZ003). We also thank the HPC platform
of Xi’An Jiaotong University, where our numerical calculations was performed.

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Supplementary Material

##  S-0.1 Experimental implementation

Our experimental setup is sketched in Fig. 2. Quantum walks are implemented
via the temporal-mode multiplexing scheme. The photon’s wave packet is
provided by a pulsed laser source with central wavelength of $808$ $nm$, pulse
width of $64$ $ps$, and repetition rate of $31.25$ $kHz$. The pulses are
attenuated to the single-photon level by using neutral density filters,
ensuring a negligible probability of multi-photon events Schreiber et al.
(2010, 2012). The polarization-dependent translation operator is realized in
the temporal domain through building the path-dependent time delays into two
different paths for horizontally and vertically polarized photons,
respectively. We experimentally realize a polarization selective-loss operator
$M_{p}=\mathbb{I}_{x}\otimes(e^{2\gamma}\left|\downarrow\right\rangle\left\langle\downarrow\right|+\left|\uparrow\right\rangle\left\langle\uparrow\right|)$
through adding an attenuator to the path of horizontally polarized photons.
Since $M=e^{-\gamma}M_{p}$, it is straightforward to map the experimentally
implemented dynamics to those under $U_{0}$ by multiplying a time dependent
factor $e^{-\gamma t}$. The coin states are encoded in the photon
polarizations and the corresponding coin operator is implemented through the
half-wave plates (HWPs), which can provide a careful control over the
parameters $\theta_{1}$ and $\theta_{2}$. The probability distribution of the
walker is measured by employing the avalanche photo-diode (APD) to record the
information regarding the number of time steps as well as the number of
photons.

##  S-0.2  The split-step quantum walk

Our studied non-chiral non-unitary quantum walk in a 1D lattice is implemented
through the so-called split-step quantum walk scheme Kitagawa et al. (2012,
2010), which can be described through the following unitary evolution operator
$U_{0}=T_{\downarrow}R_{y}(\theta_{2})MT_{\uparrow}R_{y}(\theta_{1})$.
$R_{y}(\theta)\equiv\mathbb{I}_{x}\otimes e^{-i\theta\sigma_{y}/2}$ represents
the rotation of the coin state about the y axis, with
$\mathbb{I}_{x}\equiv\sum\limits_{j}\left|j\right\rangle\left\langle j\right|$
being the identity matrix in lattice modes. The pseudospin-dependent
translation by one lattice site is captured by
$T_{\uparrow}$($T_{\downarrow}$) defined as follows:

$\displaystyle\begin{split}T_{\uparrow}&\equiv\sum\limits_{j}\left|j\right\rangle\left\langle
j-1\right|\otimes\left|\uparrow\right\rangle\left\langle\uparrow\right|+\mathbb{I}_{x}\otimes\left|\downarrow\right\rangle\left\langle\downarrow\right|\\\
T_{\downarrow}&\equiv\sum\limits_{j}\left|j\right\rangle\left\langle
j+1\right|\otimes\left|\downarrow\right\rangle\left\langle\downarrow\right|+\mathbb{I}_{x}\otimes\left|\uparrow\right\rangle\left\langle\uparrow\right|\end{split}$
(S1)

where $j$ is the site index of 1D lattice and $\downarrow$ and $\uparrow$
represent the coin state $\left|0\right\rangle$ and $\left|1\right\rangle$,
respectively. $M$ is the polarization selective-loss operator with $\gamma$
being a tunable parameter in experiments.

##  S-0.3 The bulk of the non-chiral non-unitary quantum walk

To investigate the bulk of our 1D non-chiral non-unitary quantum walk, we
rewrite the bulk eigenstate $\left|\psi\right\rangle$ as

$\left|\psi\right\rangle=\sum\limits_{j,n}\beta_{n}^{j}\left|j\right\rangle\otimes\left|\phi_{n}\right\rangle$
(S2)

where ${\beta_{n}}$ is the spatial-mode function for the $n$-th mode, which
takes into account the generic deviations of localized bulk states from Bloch
waves. ${\phi_{n}}$ is the corresponding coin state and $j$ is the site index
of 1D lattice. The bulk of the 1D chain can be investigated through solving
the following eigenstate equation

$(A_{m}\beta_{n}+A_{p}\beta_{n}^{-1}+A_{s}-\lambda)\left|\phi_{n}\right\rangle=0$
(S3)

where $A_{m}=F_{m}MG_{s}$ , $A_{p}=F_{s}MG_{p}$ and
$A_{s}=F_{s}MG_{s}+F_{m}MG_{p}$ with $F_{m}=P_{\downarrow}R_{y}(\theta_{2})$ ,
$F_{s}=P_{\uparrow}R_{y}(\theta_{2})$ ,
$G_{s}=P_{{}_{\downarrow}}R_{y}(\theta_{1})$ and
$G_{p}=P_{\uparrow}R_{y}(\theta_{1})$. Here
$P_{\uparrow}=\left|\uparrow\right\rangle\left\langle\uparrow\right|$,
$P_{{}_{\downarrow}}=\left|\downarrow\right\rangle\left\langle\downarrow\right|$
and $\lambda=\exp(-iE)$. To make the above equation have non-trivial
solutions, the following relation should be satisfied

$\det(A_{m}\beta_{n}+A_{p}\beta_{n}^{-1}+A_{s}-\lambda)=0$ (S4)

Through solving Eq. (S3), we find that there exist two solutions of
${\beta_{n}}$ with $n=1,2$ and Eq. (S3) can thus be rewritten as

$(A_{m}\beta_{1}+A_{p}\beta_{1}^{-1}+A_{s}-\lambda)\left|\phi_{1}\right\rangle=0$

$(A_{m}\beta_{2}+A_{p}\beta_{2}^{-1}+A_{s}-\lambda)\left|\phi_{2}\right\rangle=0$
(S5)

The energy spectrum can thus be obtained as

$\displaystyle\begin{split}E\equiv
E_{\pm}&=\pm\arccos(-\frac{\alpha+\alpha^{-1}}{2}\sin\theta_{1}/2\sin\theta_{2}/2+\frac{\beta_{n}\alpha+\beta_{n}^{-1}\alpha^{-1}}{2}\cos\theta_{1}/2\cos\theta_{2}/2)\end{split}$
(S6)

The effective non-Hermitian Hamiltonian constructed from the above equation
Eq. (S4) can be expressed in the form $\mathbf{h\cdot\sigma}$

$\displaystyle\begin{split}h_{x}&=-\frac{E}{2i\sin
E}(\beta_{n}a-\beta_{n}^{-1}a^{-1})\cos\theta_{2}/2\\\ h_{y}&=-\frac{E}{2i\sin
E}[i(a^{-1}+a)\cos\theta_{1}/2\sin\theta_{2}/2+i(\beta_{n}a+\beta_{n}^{-1}a^{-1})\sin\theta_{1}/2\cos\theta_{2}/2]\\\
h_{z}&=\frac{E}{2i\sin E}(a-a^{-1})\sin\theta_{2}/2\end{split}$ (S7)

The corresponding eigenstates can be obtained

$\left|\psi_{\pm}^{R}\right\rangle=\frac{[\sin E_{\pm}(E_{\pm}-\sin
2E_{\pm}h_{y})+\sin E_{\pm}^{\ast}(E_{\pm}+\sin
2E_{\pm}h_{y})]^{1/2}}{E_{\pm}\sqrt{2(E_{\pm}-h_{z})}}(h_{x}-ih_{y},E_{\pm}-h_{z})^{T}$
(S8) $\left\langle\psi_{\pm}^{L}\right|=\frac{[\sin E_{\pm}(E_{\pm}-\sin
2E_{\pm}h_{y})+\sin E_{\pm}^{\ast}(E_{\pm}+\sin
2E_{\pm}h_{y})]^{-1/2}}{\sqrt{2(E_{\pm}-h_{z})}}(h_{x}+ih_{y},E_{\pm}-h_{z})$
(S9)

Since the OBC imposes one further condition
$\left|{{\beta_{1}}}\right|=\left|{{\beta_{2}}}\right|$ for the bulk modes,
these spatial modes can be labeled by $\beta$, which will form the GBZ as
shown in the red circle in Fig. 1(c). Through Eq. (S6), we can obtain the
effective non-Hermitian Hamiltonian ${H_{eff}}$ as shown in Eq. (3) in the
main text.

##  S-0.4 Constructing GBZ under the generalized boundary condition (GBC)

To construct our proposed non-chiral non-Bloch topological invariants, we
consider imposing the generalized boundary condition (GBC) to our system,
which can be captured by the following Floquet operator
$U=T_{\downarrow}^{\prime}R_{y}(\theta_{2})MT_{\uparrow}^{\prime}R_{y}(\theta_{1})$.
To find out the generalized Brillouin zone (GBZ) under GBC, i.e., considering
$U$ with a certain novanished $\Delta$ , we diagonalize $U$ in its real space
matrix form $\left(\begin{array}[c]{cc}U_{i,j}^{11}&U_{i,j}^{12}\\\
U_{i,j}^{21}&U_{i,j}^{22}\end{array}\right)$ with
$U_{i,j}^{11}=(\delta_{i+1,j}+\delta_{i,L}\delta_{1,j}\Delta)e^{\gamma}\cos\theta_{1}/2\cos\theta_{2}/2-(\delta_{i,j}+\delta_{i,L}\delta_{L,j}\Delta^{2})e^{-\gamma}\sin\theta_{1}/2\sin\theta_{2}/2$,
$U_{i,j}^{22}=(\delta_{i,j+1}+\delta_{i,1}\delta_{L,j}\Delta)e^{-\gamma}\cos\theta_{1}/2\cos\theta_{2}/2-\delta_{i,j}e^{\gamma}\sin\theta_{1}/2\sin\theta_{2}/2$,
$U_{i,j}^{21}=(\delta_{i,j+1}+\delta_{i,1}\delta_{L,j}\Delta)e^{-\gamma}\sin\theta_{1}/2\cos\theta_{2}/2+\delta_{i,j}e^{\gamma}\cos\theta_{1}/2\sin\theta_{2}/2$
and
$U_{i,j}^{12}=-(\delta_{i+1,j}+\delta_{i,L}\delta_{1,j}\Delta)e^{\gamma}\sin\theta_{1}/2\cos\theta_{2}/2-(\delta_{i,j}+\delta_{i,L}\delta_{L,j}\Delta^{2})e^{-\gamma}\cos\theta_{1}/2\sin\theta_{2}/2$.
The continuum spectrum can thus be found out. Through substituting the
quasienergy of the continuum spectrum into Eq. (S5), the corresponding
spatial-mode function ${\beta_{n}}$ can be obtained. We thus plot these
${\beta_{n}}$ in the complex plane, as shown in Fig. 1(c) in the main text,
the GBZ under GBC can be constructed.


# Directional mean dimension and continuum-wise expansive
$\mathbb{Z}^{k}$-actions

Sebastián Donoso  Center for Mathematical Modeling and Department of
Mathematical Engineering, University of Chile and IRL 2807 - CNRS
<EMAIL_ADDRESS>,  Lei Jin  Center for Mathematical Modeling, University
of Chile and IRL 2807 - CNRS<EMAIL_ADDRESS>,  Alejandro Maass  Center for
Mathematical Modeling and Department of Mathematical Engineering, University
of Chile and IRL 2807 - CNRS<EMAIL_ADDRESS>and  Yixiao Qiao  School of
Mathematics and Statistics, Guangdong University of Technology, Guangzhou,
510006, China<EMAIL_ADDRESS>
# Directional mean dimension and continuum-wise expansive
$\mathbb{Z}^{k}$-actions

Sebastián Donoso  Center for Mathematical Modeling and Department of
Mathematical Engineering, University of Chile and IRL 2807 - CNRS
<EMAIL_ADDRESS>,  Lei Jin  Center for Mathematical Modeling, University
of Chile and IRL 2807 - CNRS<EMAIL_ADDRESS>,  Alejandro Maass  Center for
Mathematical Modeling and Department of Mathematical Engineering, University
of Chile and IRL 2807 - CNRS<EMAIL_ADDRESS>and  Yixiao Qiao  School of
Mathematics and Statistics, Guangdong University of Technology, Guangzhou,
510006, China<EMAIL_ADDRESS>
###### Abstract.

We study directional mean dimension of $\mathbb{Z}^{k}$-actions (where $k$ is
a positive integer). On the one hand, we show that there is a
$\mathbb{Z}^{2}$-action whose directional mean dimension (considered as a
$[0,+\infty]$-valued function on the torus) is not continuous. On the other
hand, we prove that if a $\mathbb{Z}^{k}$-action is continuum-wise expansive,
then the values of its $(k-1)$-dimensional directional mean dimension are
bounded. This is a generalization (with a view towards Meyerovitch and
Tsukamoto’s theorem on mean dimension and expansive multiparameter actions) of
a classical result due to Mañé: Any compact metrizable space admitting an
expansive homeomorphism (with respect to a compatible metric) is finite-
dimensional.

###### Key words and phrases:

Directional mean dimension; Continuum-wise expansive $\mathbb{Z}^{k}$-action.

###### 2010 Mathematics Subject Classification:

37B05.

S.D. was supported by Centro de Modelamiento Matemático (CMM), ACE210010 and
FB210005, BASAL funds for centers of excellence from ANID-Chile, and
ANID/Fondecyt/1200897.

L.J. was supported by Basal Funding AFB 170001 and ANID/Fondecyt/3190127, and
was partially supported by NNSF of China No. 11971455.

A.M. was supported by Centro de Modelamiento Matemático (CMM), ACE210010 and
FB210005, BASAL funds for centers of excellence from ANID-Chile.

Y.Q. was supported by NNSF of China No. 11901206.

##  1\. Introduction

The notion of expansiveness (or unstable homeomorphisms), introduced in 1950
by Utz [Utz50], is a dynamical property shared by a class of systems
exhibiting chaotic behaviors. Let $(X,d)$ be a compact metric space. A
homeomorphism $T\colon X\to X$ is said to be expansive if there is a constant
$c>0$ satisfying that for any distinct points $x,x^{\prime}\in X$ there exists
$n\in\mathbb{Z}$ such that $d(T^{n}x,T^{n}x^{\prime})>c$. In 1979, Mañé
[Man79] established a fairly surprising result which states that any compact
metrizable space admitting an expansive homeomorphism (with respect to a
compatible metric) is finite dimensional. In contrast to $\mathbb{Z}$-actions,
there do exist expansive $\mathbb{Z}^{2}$-actions on infinite dimensional
compact metric spaces [MT19, SZ05]. Such examples seem to indicate that Mañé’s
theorem cannot be extended to $\mathbb{Z}^{k}$-actions, for $k\geq 2$ or to
more general group actions. However, Meyerovitch and Tsukamoto [MT19]
succeeded in finding a reasonable framework to study extension of Mañé’s
results, with a view towards mean dimension.

Mean dimension was introduced by Gromov [Gro99] in 1999. It is a topological
invariant of dynamical systems, whose advantage has been shown excellently in
the study of infinite dimensional and infinite topological entropy systems. We
will review its definition in Subsection 2.4. It is worth mentioning that mean
dimension has deep and significant applications to dynamical systems. In
particular, it is intimately connected with the embedding problem. For related
results we refer to [GLT16, GQT19, GT20, Lin99, LT18, LT19, LW00].

Meyerovitch and Tsukamoto [MT19] provided a striking generalization of Mañé’s
theorem to $\mathbb{Z}^{k}$-actions. They showed that if
$(X,\mathbb{Z}^{k},T)$ is an expansive $\mathbb{Z}^{k}$-action (where $k$ is a
positive integer) and if a $\mathbb{Z}^{k-1}$-action $(X,\mathbb{Z}^{k-1},R)$
satisfies that $R\colon\mathbb{Z}^{k-1}\times X\to X$ commutes with
$T\colon\mathbb{Z}^{k}\times X\to X$, then $(X,\mathbb{Z}^{k-1},R)$ has finite
mean dimension. Note that this statement reduces to Mañé’s result when $k=1$,
since a trivial action has finite mean dimension if and only if the space is
finite dimensional. Moreover, they introduced the notion of directional mean
dimension [MT19], which was initially suggested by Lind, mimicking the
definition of directional entropy in [BL97]. Directional mean dimension is
able to measure the “averaged dimension” of a dynamical system along a given
subspace or direction111We shall use the word “direction” even if the subspace
is not one dimensional. We state its precise definition (for
$\mathbb{Z}^{k}$-actions) in Section 3. This is the starting point of what we
plan to investigate in the present paper. Our motivation lies mainly in two
aspects, which we explain shortly as follows.

For a $\mathbb{Z}^{2}$-action, let us consider its directional mean dimension
as a $[0,+\infty]$-valued function on the torus. An immediate and natural
problem is if such a function is continuous. We remark that if this is true,
then Meyerovitch and Tsukamoto’s theorem will imply that if a
$\mathbb{Z}^{2}$-action is expansive then its directional mean dimension along
any direction is finite (because a continuous $[0,+\infty]$-valued function on
a compact metric space is bounded by a finite real number provided it is
finite at some point). However, it turns out that we cannot expect directional
mean dimension functions to have such a strong property. We will construct a
$\mathbb{Z}^{2}$-action whose directional mean dimension function fails to be
continuous on the torus.

Let
$\mathbb{S}=\\{\vec{v}=(v_{1},v_{2})\in\mathbb{R}^{2}:v_{1}^{2}+v_{2}^{2}=1\\}$.
For a $\mathbb{Z}^{2}$-action $(X,\mathbb{Z}^{2},T)$ we denote by
$\operatorname{mdim}(X,T,\langle\vec{v}\rangle^{\perp})$ the directional mean
dimension of $(X,\mathbb{Z}^{2},T)$ with respect to the $1$-dimensional
subspace of $\mathbb{R}^{2}$ orthogonal to a direction $\vec{v}\in\mathbb{S}$.

######  Theorem 1.1.

For any $0\leq\rho\leq+\infty$ there is a $\mathbb{Z}^{2}$-action
$(X,\mathbb{Z}^{2},T)$ such that

  * •

if $\vec{v}\in\\{(-1,0),(1,0)\\}$ then
$\operatorname{mdim}(X,T,\langle\vec{v}\rangle^{\perp})=\rho$;

  * •

if $\vec{v}\in\mathbb{S}\setminus\\{(-1,0),(1,0)\\}$ then
$\operatorname{mdim}(X,T,\langle\vec{v}\rangle^{\perp})=0$.

Theorem 1.1 shows in particular that directional mean dimension need not
depend continuously on directions, but still leaves open if expansive
$\mathbb{Z}^{2}$-actions have finite directional mean dimension for all
directions. We shall address this question in a wider context. Indeed,
different versions of expansiveness were well explored previously. For
instance, there are entropy-expansiveness [Bow72], continuum-wise
expansiveness [Kat93], pointwise expansiveness [Red70] and positive
expansiveness [Sch52]. Among all the variants, continuum-wise expansiveness
attracts a lot of attention. We will recall its definition formally in
Subsection 2.2. It is worth mentioning that the class of continuum-wise
expansive homeomorphisms contains, in addition to expansive homeomorphisms,
many important homeomorphisms which are not expansive; for example, the shift
maps of Knaster’s indecomposable chainable continua [Kat93]; meanwhile, in
contrast to expansive homeomorphisms, there exist continuum-wise expansive
homeomorphisms on the pseudo-arc. Kato [Kat93] extended Mañé’s result to the
setting of continuum-wise expansive homeomorphisms. A purpose of this paper is
to prove that all continuum-wise expansive $\mathbb{Z}^{k}$-actions (in
particular, all expansive $\mathbb{Z}^{k}$-actions) must have finite
directional mean dimension.

######  Theorem 1.2.

Let $k$ be a positive integer. If a $\mathbb{Z}^{k}$-action
$(X,\mathbb{Z}^{k},T)$ is continuum-wise expansive, then the
$(k-1)$-dimensional directional mean dimension of $(X,\mathbb{Z}^{k},T)$ with
respect to any direction is finite. More precisely, they are uniformly bounded
by a finite number which depends only on $(X,\mathbb{Z}^{k},T)$.

We remark that the finiteness in the statement of Theorem 1.2 cannot be
improved to zero. In fact, it is possible to construct an expansive
$\mathbb{Z}^{2}$-action (even minimal) of positive directional mean dimension
(for details see [MT19]). When $k=1$, Theorem 1.2 coincides with Mañé’s and
Kato’s results.

This paper is organized as follows: In Section 2, we collect basic notions,
terminologies, and some known propositions in mean dimension and continuum
theory. In Section 3, we study directional mean dimension of
$\mathbb{Z}^{k}$-actions, where we shall present its fundamental properties
and where we shall provide a constructive proof of Theorem 1.1. In Section 4,
we will prove Theorem 1.2.

##  2\. Preliminaries

###  2.1. Continua

By a continuum we understand a connected compact metric space. We refer to
[Nad78, Nad92] for a systematic study of continuum theory. Let $(X,d)$ be a
compact metric space. By the hyperspace of $X$, denoted by $2^{X}$, we mean
the space of all nonempty closed subsets of $X$, equipped with the Hausdorff
metric $d_{H}(A,B)=\inf\\{\epsilon>0:B\subset U_{\epsilon}(A),A\subset
U_{\epsilon}(B)\\}$, where $U_{\epsilon}(A)$ denotes the
$\epsilon$-neighbourhood (with respect to the metric $d$ on $X$) of $A$ in
$X$. We denote by $C(X)$ the space of all nonempty connected closed subsets of
$X$, endowed with the Hausdorff metric $d_{H}$. It is well known [Nad92,
Theorems 4.13, 4.17] that if $X$ is a compact metric space, then both $2^{X}$
and $C(X)$ are compact. For a topological space $X$ the connected component of
a point $x\in X$ is the union of all the connected subsets of $X$ containing
$x$. Every connected component is connected and closed in $X$. The connected
components of the points of $X$ form a partition of $X$. When concerning a
space the term nondegenerate will always mean that the space consists of at
least two points.

###  2.2. Continuum-wise expansive $\mathbb{Z}^{k}$-actions

Let $k$ be a positive integer. We say that a triple $(X,\mathbb{Z}^{k},T)$ is
a $\mathbb{Z}^{k}$-action if $X$ is a compact metric space and
$T\colon\mathbb{Z}^{k}\times X\to X$, $(n,x)\mapsto T^{n}x$ is a continuous
mapping satisfying $T^{0}x=x$ and $T^{m+n}(x)=T^{m}(T^{n}(x))$ for all
$m,n\in\mathbb{Z}^{k}$ and all $x\in X$. A $\mathbb{Z}^{k}$-action
$(X,\mathbb{Z}^{k},T)$ is said to be continuum-wise expansive if there is a
constant $c>0$ satisfying that for every nondegenerate $A\in C(X)$ there
exists $n\in\mathbb{Z}^{k}$ such that $\operatorname{diam}T^{n}(A)>c$. Such a
constant $c>0$ is called an expansivity constant for $(X,\mathbb{Z}^{k},T)$.
Note that expansive $\mathbb{Z}^{k}$-actions are obviously continuum-wise
expansive.

###  2.3. Topological dimension

Let $X$ be a compact metric space. The join of open covers $\mathcal{U}$ and
$\mathcal{V}$ of $X$ is the open cover
$\mathcal{U}\vee\mathcal{V}\coloneqq\\{U\cap
V:U\in\mathcal{U},V\in\mathcal{V}\\}$. For a cover $\mathcal{U}$ of $X$ we set
$\operatorname{mesh}(\mathcal{U},d)=\sup_{U\in\mathcal{U}}\operatorname{diam}(U)$.
A cover $\mathcal{V}$ refines $\mathcal{U}$ if for every $V\in\mathcal{V}$
there exists $U\in\mathcal{U}$ with $V\subset U$. Let
$\mathcal{U}=\\{U_{i}\\}_{i=1}^{n}$ be a finite cover of $X$. We define the
order of $\mathcal{U}$ by $\operatorname{ord}(\mathcal{U})=\max_{x\in
X}\sum_{i=1}^{n}1_{U_{i}}(x)-1$ and the degree of $\mathcal{U}$ by
$\mathcal{D}(\mathcal{U})=\min_{\mathcal{V}}\operatorname{ord}(\mathcal{V})$,
where $\mathcal{V}$ runs over all finite open covers of $X$ refining
$\mathcal{U}$. The topological dimension of $X$ is defined by
$\dim(X)=\sup_{\mathcal{U}}\mathcal{D}(\mathcal{U})$, where $\mathcal{U}$ runs
over all finite open covers of $X$. As follows we list some elementary
propositions about $\mathcal{D}$, which will be used in this paper.

######  Lemma 2.1 ([Coo15, Propositions 1.6.5, 4.4.5]).

Let $X$ be a compact metric space and $\alpha$ a finite open cover of $X$.
Then the following statements hold:

  1. (1)

$\mathcal{D}(\alpha)=\min_{\beta}\operatorname{ord}(\beta)$, where $\beta$
runs over all finite closed covers of $X$ refining $\alpha$;

  2. (2)

$\mathcal{D}(\alpha)\leq l$ if and only if there is a compact metric space $P$
of topological dimension $l$ and a continuous mapping $f\colon X\to P$ which
is $\alpha$-compatible (i.e. there exists a finite open cover $\beta$ of $P$
such that $f^{-1}(\beta)$ refines $\alpha$).

######  Lemma 2.2 ([LW00, Corollary 2.5]).

Suppose that $\mathcal{U}$ and $\mathcal{V}$ are finite open covers of a
compact metric space $X$. Then we have
$\mathcal{D}(\mathcal{U}\vee\mathcal{V})\leq\mathcal{D}(\mathcal{U})+\mathcal{D}(\mathcal{V})$.

Let $(X,d)$ be a compact metric space, $Y$ a topological space, and
$\epsilon>0$. A continuous mapping $f\colon X\to Y$ is called an
$\epsilon$-embedding if $f(x)=f(x^{\prime})$ implies
$d(x,x^{\prime})<\epsilon$ for all $x,x^{\prime}\in X$. We define
$\operatorname{Widim}_{\epsilon}(X,d)=\min_{K}\dim(K)$, where $K$ ranges over
all compact metric spaces admitting an $\epsilon$-embedding $f\colon X\to K$.
Here we put a compatible metric $d$ on $X$ explicitly in the notation
$\operatorname{Widim}_{\epsilon}(X,d)$ because
$\operatorname{Widim}_{\epsilon}(X,d)$ does depend on $d$. Nevertheless, it is
well known [Coo15, Chapter 4], for any compatible metric $d$ on $X$, that
$\dim(X)=\lim_{\epsilon\to 0}\operatorname{Widim}_{\epsilon}(X,d)$.

###  2.4. Mean dimension

Let $k$ and $N$ be positive integers. We denote

$[-N,N]^{k}=\\{(n_{i})_{i=1}^{k}\in\mathbb{Z}^{k}:-N\leq n_{i}\leq N,1\leq
i\leq k\\},$
$\partial[-N,N]^{k}=\bigcup_{j=1}^{k}\\{(x_{i})_{i=1}^{k}\in[-N,N]^{k}:x_{j}\in\\{-N,N\\}\\}.$

We caution the readers that the notation $[-N,N]^{k}$ above depends on the
context. In some places in the text, $[-N,N]$ may mean a real interval, but we
believe that this will not cause any confusion at all. Let
$(X,\mathbb{Z}^{k},T)$ be a $\mathbb{Z}^{k}$-action. We set for a finite open
cover $\mathcal{U}$ of $X$

(2.1)
$\operatorname{mdim}(X,T;\mathcal{U})=\lim_{N\to+\infty}\frac{\mathcal{D}(\bigvee_{n\in[-N,N]^{k}}T^{n}\mathcal{U})}{(2N+1)^{k}}$

and define the mean dimension of $(X,\mathbb{Z}^{k},T)$ by

$\operatorname{mdim}(X,\mathbb{Z}^{k},T)=\sup_{\mathcal{U}}\operatorname{mdim}(X,T;\mathcal{U}),$

where $\mathcal{U}$ ranges over all finite open covers of $X$. The limit in
(2.1) always exists due to Lemma 2.2. Clearly,
$\operatorname{mdim}(X,\mathbb{Z}^{k},T)\in[0,+\infty]$. An easy observation
is that if $\dim(X)<+\infty$ then $\operatorname{mdim}(X,\mathbb{Z}^{k},T)=0$.
Moreover, it was shown [LW00, Theorem 4.2] that if $(X,\mathbb{Z}^{k},T)$ has
finite topological entropy, then $\operatorname{mdim}(X,\mathbb{Z}^{k},T)=0$.
Typical examples of dynamical systems having positive mean dimension are full
shifts (over a positive dimension alphabet). Let $D$ be a positive integer.
The full shift on $([0,1]^{D})^{\mathbb{Z}^{k}}$ is defined by
$\sigma\colon\mathbb{Z}^{k}\times([0,1]^{D})^{\mathbb{Z}^{k}}\to([0,1]^{D})^{\mathbb{Z}^{k}}$,
$(n,(x_{m})_{m\in{\mathbb{Z}^{k}}})\mapsto(x_{m+n})_{m\in{\mathbb{Z}^{k}}}$.
It was proved in [LW00] that
$\operatorname{mdim}(([0,1]^{D})^{\mathbb{Z}^{k}},\sigma)=D$. Let $d$ be a
compatible metric on $X$. For any finite subset $\Omega$ of $\mathbb{Z}^{k}$
we define $d_{\Omega}^{T}$ by

$d_{\Omega}^{T}(x,x^{\prime})=\max_{n\in\Omega}d(T^{n}x,T^{n}x^{\prime}),\quad\text{for
}x,x^{\prime}\in X.$

It is clear that $d_{\Omega}^{T}$ is also a compatible metric on $X$.
Furthermore, we can verify that

$\operatorname{mdim}(X,\mathbb{Z}^{k},T)=\lim_{\epsilon\to
0}\left(\lim_{N\to+\infty}\frac{\operatorname{Widim}_{\epsilon}(X,d_{[-N,N]^{k}}^{T})}{(2N+1)^{k}}\right).$

##  3\. Directional mean dimension

Let $k\in\mathbb{N}$ and fix a real number $r>\sqrt{k}/2$. Let
$V\subseteq\mathbb{R}^{k}$ be an $h$-dimensional subspace of $\mathbb{R}^{k}$.
Define $B_{r}(V)\coloneqq\\{u\in\mathbb{Z}^{k}:|u-w|<r\text{ for some }w\in
V\\}$. For a $\mathbb{Z}^{k}$-action $(X,\mathbb{Z}^{k},T)$, with a compatible
metric $d$ on $X$, we define the $h$-dimensional directional mean dimension of
$(X,\mathbb{Z}^{k},T)$ with respect to $V$ by

(3.1) $\operatorname{mdim}(X,T,V)=\lim_{\epsilon\to
0}\left(\liminf_{N\to+\infty}\frac{\operatorname{Widim}_{\epsilon}(X,d_{B_{r}(V)\cap[-N,N]^{k}}^{T})}{\operatorname{vol}_{h}(V\cap[-N,N]^{k})}\right),$

where $\operatorname{vol}_{h}(V\cap[-N,N]^{k})$ denotes the $h$-dimensional
volume inside $V$ of its intersection with the real cube $[-N,N]^{k}$ 222It is
possible to give a much more general definition, following the notions of
directional entropy in [BL97], but for the sake of simplicity of the paper, we
do not do so here.. It follows that there exist positive constants $C_{1}$ and
$C_{2}$ that depend only on the dimension $k$ and are independent of $V$ such
that

$C_{1}N^{h}\leq\operatorname{vol}_{h}(V\cap[-N,N]^{k})\leq C_{2}N^{h}.$

We note that we shall write $|A|$ for the cardinality of a set $A$ in the
sequel.

The constant $r>\sqrt{k}/2$ is to guarantee that for every point $u\in V$
there is $v\in\mathbb{Z}^{k}$ with $|u-v|<r$, and hence $B_{r}(V)$ is always
an infinite set. We do not include $r$ in the notation
$\operatorname{mdim}(X,T,V)$ because the definition of directional mean
dimension is independent of the choices of $r>\sqrt{k}/2$ and the compatible
metrics $d$ on $X$ (see Proposition 3.1).

For each $1\leq i\leq k$ we set $\vec{e_{i}}=(0,\dots,0,1,0,\dots,0)$ (i.e.
having $1$ only on the $i$-th coordinate) and denote by
$\mathbb{Z}_{i}^{k}=\langle e_{i}\rangle^{\perp}\cap\mathbb{Z}^{k}$ the
subgroup of $\mathbb{Z}^{k}$ consisting of all the elements having zero on the
$i$-th coordinate. For a $\mathbb{Z}^{k}$-action $(X,\mathbb{Z}^{k},T)$ and
every $1\leq i\leq k$ we let $(X,\mathbb{Z}^{k-1},T|_{\mathbb{Z}_{i}^{k}})$
(which is a $\mathbb{Z}^{k-1}$-action) be the restriction of $T$ to the
subgroup $\mathbb{Z}_{i}^{k}$.

######  Proposition 3.1.

The following statements are true for all $\mathbb{Z}^{k}$-actions
$(X,\mathbb{Z}^{k},T)$:

(1) For any $1\leq h\leq k$ and any $h$-dimensional subspace
$V\subseteq\mathbb{R}^{k}$, the value $\operatorname{mdim}(X,T,V)$ is
independent of $r>\sqrt{k}/2$.

(2) Directional mean dimension does not depend on a choice of a compatible
metric on $X$; more precisely, for any $h$-dimensional subspace
$V\subseteq\mathbb{R}^{k}$ and $r>\sqrt{k}/2$

(3.2)
$\operatorname{mdim}(X,T,V)=\sup_{\mathcal{U}}\left(\liminf_{N\to+\infty}\frac{\mathcal{D}(\vee_{n\in
B_{r}(V)\cap[-N,N]^{k}}T^{n}\mathcal{U})}{\operatorname{vol}_{h}(V\cap[-N,N]^{k})}\right),$

where $\mathcal{U}$ ranges over all finite open covers of $X$.

(3) For any $1\leq i\leq k$ we have
$\operatorname{mdim}(X,T,\langle\vec{e_{i}}\rangle^{\perp})=\operatorname{mdim}(X,\mathbb{Z}^{k-1},T|_{\mathbb{Z}_{i}^{k}})$.

###### Proof.

(1) In the first part of the proof we shall use the notation
$\operatorname{mdim}(X,T,V;r)$ for the right hand side of (3.2) (instead of
$\operatorname{mdim}(X,T,V)$), since our current goal is to show that this
quantity does not depend on $r>\sqrt{k}/2$. Take $\sqrt{k}/2<r_{1}<r_{2}$. On
the one hand, since

$d_{B_{r_{1}}(V)\cap[-N,N]^{k}}^{T}(x,y)\leq
d_{B_{r_{2}}(V)\cap[-N,N]^{k}}^{T}(x,y)$

for all $x,y\in X$ and $N\in\mathbb{N}$, we have

$\operatorname{Widim}_{\epsilon}(X,d_{B_{r_{1}}(V)\cap[-N,N]^{k}}^{T})\leq\operatorname{Widim}_{\epsilon}(X,d_{B_{r_{2}}(V)\cap[-N,N]^{k}}^{T})$

for all $N\in\mathbb{N}$ and $\epsilon>0$. Thus
$\operatorname{mdim}(X,T,V;r_{1})\leq\operatorname{mdim}(X,T,V;r_{2})$. On the
other hand, there exists a finite subset $F$ of $\mathbb{Z}^{k}$ such that for
any $N\in\mathbb{N}$ the set $B_{r_{2}}(V)\cap[-N,N]^{k}$ is contained in
$\bigcup_{m\in F}\left((B_{r_{1}}(V)\cap[-N,N]^{k})+m\right)$. For a given
$\epsilon>0$ there is $\delta>0$ such that $d(x,y)<\delta$ implies
$d_{F}^{T}(x,y)<\epsilon$. It follows that for any $N\in\mathbb{N}$ we have
$\operatorname{Widim}_{\epsilon}(X,d_{B_{r_{2}}(V)\cap[-N,N]^{k}}^{T})\leq\operatorname{Widim}_{\delta}(X,d_{B_{r_{1}}(V)\cap[-N,N]^{k}}^{T})$.
This implies that
$\operatorname{mdim}(X,T,V;r_{2})\leq\operatorname{mdim}(X,T,V;r_{1})$.

(2) To show

$\sup_{\mathcal{U}}\left(\liminf_{N\to+\infty}\frac{\mathcal{D}(\vee_{n\in
B_{r}(V)\cap[-N,N]^{k}}T^{n}\mathcal{U})}{\operatorname{vol}_{h}(V\cap[-N,N]^{k})}\right)\leq\operatorname{mdim}(X,T,\vec{v}),$

we take a finite open cover $\mathcal{U}$ of $X$ and let $\lambda>0$ be a
Lebesgue number of $\mathcal{U}$ with respect to a compatible metric $d$ on
$X$. It suffices to prove for every $N\in\mathbb{N}$ that
$\mathcal{D}(\vee_{n\in
B_{r}(V)\cap[-N,N]^{k}}T^{n}\mathcal{U})\leq\operatorname{Widim}_{\lambda}(X,d_{B_{r}(V)\cap[-N,N]^{k}}^{T})$.
In fact, we take a compact metric space $P$ with
$\dim(P)=\operatorname{Widim}_{\lambda}(X,d_{B_{r}(V)\cap[-N,N]^{k}}^{T})$ and
a $\lambda$-embedding $f\colon X\to P$ with respect to
$d_{B_{r}(V)\cap[-N,N]^{k}}^{T}$. Then it follows that
$\operatorname{diam}T^{n}(f^{-1}(p))<\lambda$ and hence $T^{n}(f^{-1}(p))$ is
contained in some element in $\mathcal{U}$ for every $n\in
B_{r}(V)\cap[-N,N]^{k}$ and every $p\in P$. Thus, $f^{-1}(p)$ is contained in
some element in $\bigvee_{n\in B_{r}(V)\cap[-N,N]^{k}}T^{n}\mathcal{U}$ for
every $p\in P$. For every $U\in\bigvee_{n\in
B_{r}(V)\cap[-N,N]^{k}}T^{n}\mathcal{U}$ we set $\tilde{U}=\\{p\in
P:f^{-1}(p)\subset U\\}$. Clearly, $f^{-1}(\tilde{U})\subset U$, the sets
$\tilde{U}$’s cover $P$, and each $\tilde{U}$ is open in $P$. So $f$ is
$\bigvee_{n\in B_{r}(V)\cap[-N,N]^{k}}T^{n}\mathcal{U}$-compatible, and
therefore by Lemma 2.1 we see that $\mathcal{D}(\vee_{n\in
B_{r}(V)\cap[-N,N]^{k}}T^{n}\mathcal{U})\leq\dim(P)$.

To show

$\operatorname{mdim}(X,T,V)\leq\sup_{\mathcal{U}}\left(\liminf_{N\to+\infty}\frac{\mathcal{D}(\vee_{n\in
B_{r}(V)\cap[-N,N]^{k}}T^{n}\mathcal{U})}{\operatorname{vol}_{h}(V\cap[-N,N]^{k})}\right),$

we let $\epsilon>0$, $N\in\mathbb{N}$, and take a finite open cover
$\mathcal{U}$ of $X$ with $\operatorname{mesh}(\mathcal{U},d)<\epsilon$. By
Lemma 2.1, we can find a compact metric space $P$ with

$\dim(P)=\mathcal{D}(\vee_{n\in B_{r}(V)\cap[-N,N]^{k}}T^{n}\mathcal{U})$

and a continuous $f\colon X\to P$ which is $\bigvee_{n\in
B_{r}(V)\cap[-N,N]^{k}}T^{n}\mathcal{U}$-compatible. So we have
$\operatorname{Widim}_{\epsilon}(X,d_{B_{r}(V)\cap[-N,N]^{k}}^{T})\leq\dim(P)$,
as required.

Statement (3) follows directly from definition. ∎

In what follows we are mainly interested in $(k-1)$-dimensional subspaces of
$\mathbb{Z}^{k}$. Note that via the orthogonal complement, these subspaces can
be parametrized with a single vector of $\mathbb{R}^{k}$. The next result
shows that $\mathbb{Z}^{k}$-actions of positive mean dimension must have
infinite directional mean dimension with respect to each
$\langle\vec{e_{i}}\rangle^{\perp}$ ($1\leq i\leq k$).

######  Proposition 3.2.

If a $\mathbb{Z}^{k}$-action $(X,\mathbb{Z}^{k},T)$ satisfies $\min_{1\leq
i\leq k}\operatorname{mdim}(X,T,\langle\vec{e_{i}}\rangle^{\perp})<+\infty$
then $\operatorname{mdim}(X,\mathbb{Z}^{k},T)=0$.

###### Proof.

Without loss of generality we assume
$\operatorname{mdim}(X,T,\langle\vec{e_{k}}\rangle^{\perp})<+\infty$. For any
positive integers $N,M$ and $j$ we put
$F(N,M;j)=[-jN,jN]^{k-1}\times[-jM,jM]$. We set
$\mathcal{F}=\\{F(N,M;1):N,M\geq 1\\}$. Clearly,
$\\{F(N,M;j)\\}_{j=1}^{\infty}$ is a Følner sequence of $\mathbb{Z}^{k}$, for
any positive integers $N$ and $M$; and $\mathcal{F}=\bigcup_{N,M\geq
1}\\{F(N,M;j)\\}_{j=1}^{\infty}$. We rewrite the definition of
$\operatorname{mdim}(X,T;\mathcal{U})$ using $\\{F(N,M;j)\\}_{j=1}^{\infty}$
in (2.1). By subadditivity of $\mathcal{D}$ we have for any finite open cover
$\mathcal{U}$ of $X$ and any $N,M\geq 1$

$\operatorname{mdim}(X,T;\mathcal{U})=\lim_{j\to+\infty}\frac{\mathcal{D}(\bigvee_{n\in
F(N,M;j)}T^{n}\mathcal{U})}{|F(N,M;j)|}=\inf_{j\geq
1}\frac{\mathcal{D}(\bigvee_{n\in F(N,M;j)}T^{n}\mathcal{U})}{|F(N,M;j)|}.$

Thus,

$\displaystyle\operatorname{mdim}(X,T;\mathcal{U})$
$\displaystyle=\inf_{N,M,j\geq 1}\frac{\mathcal{D}(\bigvee_{n\in
F(N,M;j)}T^{n}\mathcal{U})}{|F(N,M;j)|}=\inf_{F\in\mathcal{F}}\frac{\mathcal{D}(\bigvee_{n\in
F}T^{n}\mathcal{U})}{|F|}$ $\displaystyle=\inf_{N,M\geq
1}\frac{1}{2M+1}\cdot\frac{\mathcal{D}(\bigvee_{n\in[-N,N]^{k-1}}(T|_{\mathbb{Z}^{k}_{k}})^{n}(\bigvee_{m\in[-M,M]}T^{m\vec{e_{k}}}\mathcal{U}))}{(2N+1)^{k-1}}$
$\displaystyle=\inf_{M\geq 1}\frac{1}{2M+1}\cdot\inf_{N\geq
1}\frac{\mathcal{D}(\bigvee_{n\in[-N,N]^{k-1}}(T|_{\mathbb{Z}^{k}_{k}})^{n}(\bigvee_{m\in[-M,M]}T^{m\vec{e_{k}}}\mathcal{U}))}{(2N+1)^{k-1}}.$

Since $\bigvee_{m\in[-M,M]}T^{m\vec{e_{k}}}\mathcal{U}$ is independent of $N$,
we have

$\operatorname{mdim}(X,T;\mathcal{U})\leq\inf_{M\geq
1}\frac{\operatorname{mdim}(X,\mathbb{Z}^{k-1},T|_{\mathbb{Z}^{k}_{k}})}{2M+1}=\inf_{M\geq
1}\frac{\operatorname{mdim}(X,T,\langle\vec{e_{k}}\rangle^{\perp})}{2M+1}.$

Since $\operatorname{mdim}(X,T,\langle\vec{e_{k}}\rangle^{\perp})$ is finite
and since $\mathcal{U}$ is an arbitrary finite open cover of $X$, we conclude
that $\operatorname{mdim}(X,\mathbb{Z}^{k},T)=0$. ∎

We note that for a $\mathbb{Z}^{k}$-action it is possible for us to consider
its $(k-1)$-dimensional directional mean dimension as the $[0,+\infty]$-valued
function on the $(k-1)$-sphere
$\mathbb{S}^{k-1}=\\{\vec{v}\in\mathbb{R}^{k-1}:\|\vec{v}\|=1\\}$ by
associating to $\vec{v}\in\mathbb{S}^{k-1}$ the value of
$\operatorname{mdim}(X,T,\langle\vec{v}\rangle^{\perp})$. The next theorem
reveals this function is not necessarily continuous on $\mathbb{S}^{k-1}$ even
for $k=2$.

######  Theorem 3.3 (=Theorem 1.1).

For any $\rho\in[0,+\infty]$ there is a $\mathbb{Z}^{2}$-action
$(X,\mathbb{Z}^{2},T)$ such that if $\vec{v}\in\\{(-1,0),(1,0)\\}$ then
$\operatorname{mdim}(X,T,\langle\vec{v}\rangle^{\perp})=\rho$; if
$\vec{v}\in\mathbb{S}\setminus\\{(-1,0),(1,0)\\}$ then
$\operatorname{mdim}(X,T,\langle\vec{v}\rangle^{\perp})=0$. In particular, if
$\rho>0$, the $(k-1)$-dimensional directional mean dimension of
$(X,\mathbb{Z}^{2},T)$ is not continuous on $\mathbb{S}$.

###### Proof.

We provide a constructive proof, following the idea of Example 6.6 in [BL97].
The case $\rho=0$ is trivial. We can suppose $\rho=1$. Indeed, we may replace
(in the following argument) $[0,1]^{\mathbb{Z}}$ by
$([0,1]^{\mathbb{N}})^{\mathbb{Z}}$ if $\rho=+\infty$; or by a subshift (even
minimal [LW00]) of some $([0,1]^{d})^{\mathbb{Z}}$ (where $d>\rho$), whose
mean dimension is equal to $\rho$, if $0<\rho<+\infty$. So we assume $\rho=1$
for simplicity.

We let $\sigma$ be the full shift on $[0,1]^{\mathbb{Z}}$,
$\sigma\colon\mathbb{Z}\times[0,1]^{\mathbb{Z}}\to[0,1]^{\mathbb{Z}}$,
$(n,(x_{m})_{m\in\mathbb{Z}})\mapsto(x_{m+n})_{m\in\mathbb{Z}}$. We set
$Y=[0,1]^{\mathbb{Z}}\times\mathbb{Z}$. We consider an action on $Y$ as
follows:

(3.3) $\mathbb{Z}^{2}\times Y\to
Y,\quad(m,n)(x,i)=(\sigma^{m}x,n+i),\quad\forall m,n,i\in\mathbb{Z},\,\forall
x\in[0,1]^{\mathbb{Z}}.$

Let $X=Y\cup\\{\infty\\}$ be the one-point compactification of $Y$. By
defining $(m,n)\infty=\infty$ for all $(m,n)\in\mathbb{Z}^{2}$ (i.e. $\infty$
is a fixed point) we may extend the action (3.3) of $\mathbb{Z}^{2}$ on $Y$
continuously to a $\mathbb{Z}^{2}$-action $T$ on $X$.

In the remaining part of the proof we shall compute the directional mean
dimension of $(X,\mathbb{Z}^{2},T)$. We fix an arbitrary
$\vec{v}=(v_{1},v_{2})\in\mathbb{S}$.

Case 1. Suppose $v_{2}\neq 0$. We are going to prove
$\operatorname{mdim}(X,T,\langle\vec{v}\rangle^{\perp})=0$.

In fact, for any finite open cover $\mathcal{U}$ of $X$ there exist
$M\in\mathbb{N}$ sufficiently large and a finite open cover $\alpha$ of
$[0,1]^{\mathbb{Z}}$ such that $\mathcal{V}$ refines $\mathcal{U}$, where
$\mathcal{V}$ is a finite open cover of $X$ consisting of the following
members:

$\displaystyle\bullet\quad\quad A\times\\{i\\}\subset X,\quad\forall
A\in\alpha,\,\forall i\in[-M,M],$
$\displaystyle\bullet\quad\quad[0,1]^{\mathbb{Z}}\times\left(\mathbb{Z}\setminus[-M,M]\right)\cup\\{\infty\\}\subset
X.$

Hence $T^{(m,n)}\mathcal{V}$ ($(m,n)\in\mathbb{Z}^{2}$) consists of the
following members:

$\displaystyle\bullet\quad\quad\sigma^{m}(A)\times\\{i+n\\},\quad\forall
A\in\alpha,\,\forall i\in[-M,M],$
$\displaystyle\bullet\quad\quad[0,1]^{\mathbb{Z}}\times\left(\mathbb{Z}\setminus[n-M,n+M]\right)\cup\\{\infty\\}.$

We thus observe for any $(m,n)\in\mathbb{Z}^{2}$ with $n\geq 2M+1$ that
$\mathcal{V}\vee T^{(m,n)}\mathcal{V}$ consists of the following members:

$\displaystyle\bullet\quad\quad[0,1]^{\mathbb{Z}}\times\left(\mathbb{Z}\setminus([-M,M]\cup[n-M,n+M])\right)\cup\\{\infty\\},$
$\displaystyle\bullet\quad\quad A\times\\{i\\},\quad\forall
A\in\alpha,\,\forall i\in[-M,M],$
$\displaystyle\bullet\quad\quad\sigma^{m}(A)\times\\{i+n\\},\quad\forall
A\in\alpha,\,\forall i\in[-M,M].$

We note that any two elements coming from distinct lines (in the above three)
are disjoint. So we have
$\mathcal{D}(\mathcal{V})\leq\mathcal{D}(\mathcal{V}\vee
T^{(m,n)}\mathcal{V})\leq\max\\{\mathcal{D}(\mathcal{V}),\mathcal{D}(T^{(m,n)}\mathcal{V})\\}=\mathcal{D}(\mathcal{V})$.
Similarly, for any $(m,n)\in\mathbb{Z}^{2}$ with $n\leq-2M-1$ we also have
$\mathcal{D}(\mathcal{V}\vee T^{(m,n)}\mathcal{V})=\mathcal{D}(\mathcal{V})$.
Therefore, by induction we deduce that if a finite subset $F$ of
$\mathbb{Z}^{2}$ satisfies that $|q_{1}-q_{2}|\geq 2M+1$ whenever
$(p_{1},q_{1})\neq(p_{2},q_{2})\in F$, then $\mathcal{D}(\bigvee_{(p,q)\in
F}T^{(p,q)}\mathcal{V})=\mathcal{D}(\mathcal{V})$. Thus for any
$N\in\mathbb{N}$ we have

$\mathcal{D}(\bigvee_{(m,n)\in
B_{1}(\langle\vec{v}\rangle^{\perp})\cap[-N,N]^{2}}T^{(m,n)}\mathcal{V})\leq\left|B_{1}(\langle\vec{v}\rangle^{\perp})\cap[-M-1,M+1]^{2}\right|\cdot\mathcal{D}(\mathcal{V}).$

Notice that we are using $r=1>\sqrt{2}/2$ in the definition of directional
mean dimension. It follows that,

$\displaystyle\operatorname{mdim}(X,T,\langle\vec{v}\rangle^{\perp})$
$\displaystyle=\sup_{\mathcal{U}}\liminf_{N\to+\infty}\frac{\mathcal{D}(\bigvee_{(m,n)\in
B_{1}(\langle\vec{v}\rangle^{\perp})\cap[-N,N]^{2}}T^{(m,n)}\mathcal{U})}{\operatorname{vol}_{1}(\langle\vec{v}\rangle^{\perp}\cap[-N,N]^{2})}$
$\displaystyle\leq\sup_{\mathcal{U}}\liminf_{N\to+\infty}\frac{\mathcal{D}(\bigvee_{(m,n)\in
B_{1}(\langle\vec{v}\rangle^{\perp})\cap[-N,N]^{2}}T^{(m,n)}\mathcal{V})}{\operatorname{vol}_{1}(\langle\vec{v}\rangle^{\perp}\cap[-N,N]^{2})}$
$\displaystyle\leq\sup_{\mathcal{U}}\liminf_{N\to+\infty}\frac{\left|B_{1}(\langle\vec{v}\rangle^{\perp})\cap[-M-1,M+1]^{2}\right|\cdot\mathcal{D}(\mathcal{V})}{\operatorname{vol}_{1}(\langle\vec{v}\rangle^{\perp}\cap[-N,N]^{2})}$
$\displaystyle=0.$

Case 2. Suppose $v_{2}=0$. We are going to show
$\operatorname{mdim}(X,T,\langle\vec{v}\rangle^{\perp})=1$.

In fact, by Proposition 3.1.$(3)$, we have

$\operatorname{mdim}(X,T,\langle\vec{v}\rangle^{\perp})=\sup_{\alpha,M}\liminf_{N\to+\infty}\frac{\mathcal{D}(\bigvee_{n\in[-N,N]}T^{(n,0)}\mathcal{U}_{\alpha,M})}{2N+1},$

where $\mathcal{U}_{\alpha,M}$ is the finite open cover of $X$ consisting of
the following members:

$\displaystyle\bullet\quad\quad A\times\\{i\\},\quad\forall
A\in\alpha,\,\forall i\in[-M,M],$
$\displaystyle\bullet\quad\quad[0,1]^{\mathbb{Z}}\times\left(\mathbb{Z}\setminus[-M,M]\right)\cup\\{\infty\\},$

and where $\alpha$ ranges over finite open covers of $[0,1]^{\mathbb{Z}}$ and
$M$ runs over $\mathbb{N}$. Meanwhile, it is clear that
$\mathcal{D}(\bigvee_{n\in[-N,N]}T^{(n,0)}\mathcal{U}_{\alpha,M})=\mathcal{D}(\bigvee_{n\in[-N,N]}\sigma^{n}\alpha)$.
Thus we deduce

$\operatorname{mdim}(X,T,\langle\vec{v}\rangle^{\perp})=\sup_{\alpha}\liminf_{N\to+\infty}\frac{\mathcal{D}(\bigvee_{n\in[-N,N]}\sigma^{n}\alpha)}{2N+1}=\operatorname{mdim}([0,1]^{\mathbb{Z}},\sigma)=1.$

∎

##  4\. Proof of Theorem 1.2

######  Lemma 4.1 ([Nad78, Theorems 1.25, 1.26]).

For any compact metric space $X$, $A\in C(X)$ and $a\in A$ there is a
continuous mapping $\kappa:[0,1]\to C(X)$ such that $\kappa(0)=\\{a\\}$,
$\kappa(1)=A$, and $\kappa(x)\subset\kappa(y)$ for all $0\leq x\leq y\leq 1$.

######  Lemma 4.2.

Let $X$ be a compact metric space and $C$ a connected component of $X$. Then
there exists a sequence $\\{F_{n}\\}_{n=1}^{\infty}$ of clopen subsets of $X$
satisfying $F_{1}\supset F_{2}\supset\cdots\supset F_{n}\supset
F_{n+1}\supset\cdots\supset\bigcap_{n=1}^{\infty}F_{n}=C$.

###### Proof.

Without loss of generality we assume $C\neq X$. Fix $x\in C$. We denote by
$\mathcal{Q}_{x}$ the set of all clopen neighbourhoods of $x$ in $X$ and let
$Q=\bigcap_{V\in\mathcal{Q}_{x}}V$ be the quasi-component of $x$. It follows
from classical results that $C=Q$ (see for instance [Kur68, §47, Section II,
Theorem 2]; we thank the anonymous referee for pointing us to this reference)·

Since $C=Q$, for every $a\in X\setminus C$ there must be a clopen subset
$C_{a}$ of $X$ such that $C\subset C_{a}$ and $a\notin C_{a}$. This implies
that there is an open neighbourhood $U_{a}$ of $a$ in $X$ with $U_{a}\subset
X\setminus C_{a}\subset X\setminus C$. We notice that $X\setminus C$ is a
Lindelöf space and that $\\{U_{a}:a\in X\setminus C\\}$ is an open cover of
$X\setminus C$. So we can find $\\{a_{i}\\}_{i=1}^{\infty}\subset X\setminus
C$ such that $X\setminus C=\bigcup_{i=1}^{\infty}U_{a_{i}}$. Let
$F=\bigcap_{i=1}^{\infty}C_{a_{i}}$. Since for every $c\in X\setminus C$ there
is some $i\in\mathbb{N}$ with $c\in U_{a_{i}}\subset X\setminus
C_{a_{i}}\subset X\setminus F$, we see $F\subset C$. Thus, $F=C$. For each
$n\in\mathbb{N}$ we set $F_{n}=\bigcap_{i=1}^{n}C_{a_{i}}$. It is simple to
check that $\\{F_{n}\\}_{n=1}^{\infty}$ is as required. ∎

Let $(X,\mathbb{Z}^{k},T)$ be a continuum-wise expansive
$\mathbb{Z}^{k}$-action with an expansivity constant $2c>0$.

######  Lemma 4.3.

(1) There is $\delta>0$ such that if $A\in C(X)$ and $N\geq 1$ satisfy the
condition $c\leq\max\\{\operatorname{diam}T^{n}(A):n\in[-N,N]^{k}\\}\leq 2c$
then $\max\\{\operatorname{diam}T^{n}(A):n\in\partial[-N,N]^{k}\\}>\delta$.

(2) For any $\epsilon>0$ there is $m=m(\epsilon)>0$ such that if $A\in C(X)$
satisfies the condition
$\max\\{\operatorname{diam}T^{n}(A):n\in[-m,m]^{k}\\}\leq 2c$ then
$\operatorname{diam}A<\epsilon$.

###### Proof.

(1) Suppose that the statement is false. There exist
$\\{A_{i}\\}_{i\in\mathbb{N}}\subset C(X)$ and
$\\{N_{i}\\}_{i\in\mathbb{N}}\subset\mathbb{N}$ satisfying that

(4.1)
$c\leq\max\\{\operatorname{diam}T^{n}(A_{i}):n\in[-N_{i},N_{i}]^{k}\\}\leq
2c,\quad\forall i\in\mathbb{N};$ (4.2)
$\max\\{\operatorname{diam}T^{n}(A_{i}):n\in\partial[-N_{i},N_{i}]^{k}\\}\leq
1/i,\quad\forall i\in\mathbb{N}.$

Suppose that the sequence $\\{N_{i}\\}_{i\in\mathbb{N}}\subset\mathbb{N}$ is
bounded by some $0<M<+\infty$. We may assume (by the compactness of $C(X)$)
that $\\{A_{i}\\}_{i\in\mathbb{N}}$ converges to some $A\in C(X)$. Hence (4.1)
and (4.2) lead to a contradiction for $\operatorname{diam}T^{n}(A)$ for some
$n\in[-M,M]^{k}$. So we assume without loss of generality (by taking a
subsequence) that $N_{i}\to+\infty$ as $i\to\infty$.

We take a sequence $\\{n_{i}\\}_{i\in\mathbb{N}}$, with
$n_{i}\in[-N_{i},N_{i}]^{k}$ and such that
$\operatorname{diam}T^{n_{i}}(A_{i})=\max\\{\operatorname{diam}T^{n}(A_{i}):n\in[-N_{i},N_{i}]^{k}\\}$
for all $i\in\mathbb{N}$. We claim that the distance between $n_{i}$ and
$\partial[-N_{i},N_{i}]^{k}$ goes to $+\infty$ as $i\to\infty$. In fact, if
this assertion is not true, then there exists $l\in\mathbb{N}$ such that
$n_{i}\notin[-N_{i}+l,N_{i}-l]^{k}$ for infinitely many $i\in\mathbb{N}$. For
all these $i\in\mathbb{N}$ we write $n_{i}=p_{i}+q_{i}$ with
$p_{i}\in[-l,l]^{k}$ and $q_{i}\in\partial[-N_{i},N_{i}]^{k}$. We take
$\epsilon>0$ sufficiently small such that $\operatorname{diam}T^{n}(A)<c/2$
for all $n\in[-l,l]^{k}$ whenever $\operatorname{diam}A<\epsilon$. It follows
from (4.2) that
$\operatorname{diam}T^{n_{i}}(A_{i})=\operatorname{diam}T^{p_{i}}(T^{q_{i}}(A_{i}))<c/2$
for all sufficiently large $i$ (with $1/i<\epsilon$), which contradicts (4.1),
and thus proves the claim.

We assume (by choosing a subsequence) that $T^{n_{i}}(A_{i})$ tends to $B\in
C(X)$ as $i\to\infty$. For any given $m\in\mathbb{Z}^{k}$ there is some
$i_{0}\in\mathbb{N}$ such that $m+n_{i}\in[-N_{i},N_{i}]^{k}$, for all $i\geq
i_{0}$, which, together with (4.1), implies
$\operatorname{diam}T^{m+n_{i}}(A_{i})\leq\max\\{\operatorname{diam}T^{n}(A_{i}):n\in[-N_{i},N_{i}]^{k}\\}\leq
2c$. Letting $i\to\infty$ in the above inequality we obtain
$\operatorname{diam}T^{m}(B)\leq 2c$. Since $B$ is nondegenerate, this is a
contradiction.

(2) Suppose that the statement is false. There exist $\epsilon_{0}>0$ and a
sequence $\\{A_{i}\\}_{i\in\mathbb{N}}\subset C(X)$ such that
$\max\\{\operatorname{diam}T^{n}(A_{i}):n\in[-i,i]^{k}\\}\leq 2c$ and
$\operatorname{diam}A_{i}\geq\epsilon_{0}$ for all $i\in\mathbb{N}$. Choose a
subsequence of $\\{A_{i}\\}_{i\in\mathbb{N}}$ tending to some $A\in C(X)$.
Clearly, $\operatorname{diam}T^{n}(A)\leq 2c$ for all $n\in\mathbb{Z}^{k}$,
and $\operatorname{diam}A\geq\epsilon_{0}$. This is a contradiction. ∎

######  Lemma 4.4.

There is a constant $K>0$ and a sequence $\\{\mathcal{W}_{N}\\}_{N\geq 1}$ of
finite closed covers of $X$ satisfying that
$\operatorname{mesh}(\mathcal{W}_{N},d^{T}_{[-N,N]^{k}})<2c$ and
$\operatorname{ord}(\mathcal{W}_{N})\leq KN^{k-1}$, for all $N\geq 1$.

###### Proof.

Let $\delta>0$ be a constant introduced in Lemma 4.3 (1). Choose a finite open
cover $\mathcal{U}=\\{U_{1},\dots,U_{L}\\}$ of $X$ with
$\operatorname{mesh}(\mathcal{U},d)<\delta/2$. For each $N\geq 1$ we let
$\mathcal{U}_{N}$ be an open cover of $X$ defined by
$\mathcal{U}_{N}=\bigvee_{u\in\partial[-N,N]^{k}}T^{-u}\mathcal{U}$. By Lemma
2.2

(4.3)
$\mathcal{D}(\mathcal{U}_{N})\leq|\partial[-N,N]^{k}|\cdot\mathcal{D}(\mathcal{U})\leq
2^{k}(2N+1)^{k-1}L,\quad\forall N\geq 1.$

By Lemma 2.1, for each $N\geq 1$ there is a finite closed cover
$\mathcal{V}_{N}$ of $X$, which refines $\mathcal{U}_{N}$ and which satisfies
$\operatorname{ord}(\mathcal{V}_{N})=\mathcal{D}(\mathcal{U}_{N})$. For every
$V\in\mathcal{V}_{N}$ we denote by $\mathcal{C}_{V}$ the set of all connected
components of $V$. We set $\mathcal{C}_{N}=\\{C\subset
V:V\in\mathcal{V}_{N},C\in\mathcal{C}_{V}\\}$ and note that the sets in
$\mathcal{C}_{V}$ are pairwise disjoint. The set $\mathcal{C}_{N}$ is a closed
(not necessarily finite) cover of $X$, which refines $\mathcal{V}_{N}$ and
satisfies
$\operatorname{ord}(\mathcal{C}_{N})=\operatorname{ord}(\mathcal{V}_{N})$.
Claim 1. For every $N\geq 1$ we have
$\operatorname{mesh}(\mathcal{C}_{N},d^{T}_{[-N,N]^{k}})<2c$.
Proof of Claim 1. If
$\operatorname{mesh}(\mathcal{C}_{N},d^{T}_{[-N,N]^{k}})\geq 2c$ for some
$N\geq 1$, then there exists $C\in\mathcal{C}_{N}$ such that
$\max\\{\operatorname{diam}T^{n}(C):n\in[-N,N]^{k}\\}\geq 2c$ which implies
that $d^{T}_{[-N,N]^{k}}(x,y)\geq 2c$ for some $x,y\in C$. We assume
$C\in\mathcal{C}_{V}$ for some $V\in\mathcal{V}_{N}$. By Lemma 4.1, there is a
continuous mapping $\kappa\colon[0,1]\to C(V)$ satisfying $\kappa(0)=\\{x\\}$,
$\kappa(1)=C$, and $\kappa(s)\subset\kappa(t)$ whenever $0\leq s\leq t\leq 1$.
We define a continuous mapping $F\colon[0,1]\to[0,+\infty)$ by sending $t$ to
$\max\\{\operatorname{diam}T^{n}(\kappa(t)):n\in[-N,N]^{k}\\}$. Clearly,
$F(0)=0$ and $F(1)\geq 2c$. So there is some $0<t_{0}<1$ with $F(t_{0})=3c/2$.
By Lemma 4.3 (1) we get
$\max\\{\operatorname{diam}T^{n}(\kappa(t_{0})):n\in\partial[-N,N]^{k}\\}>\delta$
and hence
$\max\\{\operatorname{diam}T^{n}(C):n\in\partial[-N,N]^{k}\\}>\delta$.
However, since $\mathcal{C}_{N}$ refines $\mathcal{U}_{N}$ and since
$\operatorname{mesh}(\mathcal{U}_{N},d_{\partial[-N,N]^{k}}^{T})<\delta/2$, we
have $\max\\{\operatorname{diam}T^{n}(C):n\in\partial[-N,N]^{k}\\}<\delta/2$,
a contradiction. This shows Claim 1.
Claim 2. For every $N\geq 1$ there exists a finite closed cover
$\mathcal{W}_{N}$ of $X$ satisfying that
$\operatorname{mesh}(\mathcal{W}_{N},d^{T}_{[-N,N]^{k}})<2c$ and
$\operatorname{ord}(\mathcal{W}_{N})=\operatorname{ord}(\mathcal{V}_{N})$.
Proof of Claim 2. We fix $N\geq 1$. We take an arbitrary $V\in\mathcal{V}_{N}$
for the moment. By Claim 1,
$\operatorname{mesh}(\mathcal{C}_{V},d^{T}_{[-N,N]^{k}})<2c$. Applying Lemma
4.2 to each connected component $C$ of $V$ we find a clopen subset
$F_{C}\supset C$ of $V$, whose diameter is strictly smaller than $2c$ with
respect to $d^{T}_{[-N,N]^{k}}$. So we obtain a clopen cover
$\\{F_{C}:C\in\mathcal{C}_{V}\\}$ of $V$. Since $V$ is compact, there is a
finite subcover $\\{F_{C_{1}},F_{C_{2}},\dots,F_{C_{i(V)}}\\}$ of
$\\{F_{C}:C\in\mathcal{C}_{V}\\}$. We set $W_{1}^{V}=F_{C_{1}}$,
$W_{2}^{V}=F_{C_{2}}\setminus F_{C_{1}}$, $\dots$,
$W_{i(V)}^{V}=F_{C_{i(V)}}\setminus\cup_{j=1}^{i(V)-1}F_{C_{j}}$. It follows
that $\\{W_{1}^{V},W_{2}^{V},\dots,W_{i(V)}^{V}\\}$ is a finite closed cover
of $V$ satisfying that $W_{s}^{V}\cap W_{t}^{V}=\emptyset$ for all $1\leq
s<t\leq i(V)$. Thus, we know that $\mathcal{W}_{N}=\\{W_{j}^{V}:1\leq j\leq
i(V),V\in\mathcal{V}_{N}\\}$ is a finite closed cover of $X$. This shows Claim
2.
Now by (4.3) and noting that
$\operatorname{ord}(\mathcal{W}_{N})\leq\mathcal{D}(\mathcal{U}_{N})$ we
finish the proof of the lemma. ∎

###### Proof of Theorem 1.2.

With the above preparation we are now ready to prove Theorem 1.2. We fix a
nonzero vector $\vec{v}\in\mathbb{R}^{k}$ and $r>\frac{\sqrt{k}}{2}$. We take
$\epsilon^{\prime}>0$ arbitrarily. We choose a finite open cover $\alpha$ of
$X$, which consists of open balls of diameter smaller than $\epsilon^{\prime}$
and which has a Lebesgue number $\epsilon>0$, with respect to the distance $d$
on $X$. Note that we ensure $\epsilon<\epsilon^{\prime}$. Applying Lemma 4.3
(2) to $\epsilon>0$, there exists $m=m(\epsilon)>0$ such that for $A\in C(X)$,
if $\max\\{\operatorname{diam}T^{u+u^{\prime}}(A):u\in
B_{r}(\langle\vec{v}\rangle^{\perp})\cap[-N,N]^{k},u^{\prime}\in[-m,m]^{k}\\}\leq
2c$ then $\max\\{\operatorname{diam}T^{u}(A):u\in
B_{r}(\langle\vec{v}\rangle^{\perp})\cap[-N,N]^{k}\\}<\epsilon$. By Lemma 4.4,
there is some constant $K>0$ such that for every $N\in\mathbb{N}$ we can find
a finite closed cover $\mathcal{W}_{N+m}=\\{W_{i}\\}_{i\in I_{N}}$ of $X$
satisfying $\operatorname{mesh}(\mathcal{W}_{N+m},d^{T}_{[-N-m,N+m]^{k}})<2c$
and $\operatorname{ord}(\mathcal{W}_{N+m})<K\cdot(N+m)^{k-1}$. For each $i\in
I_{N}$ we denote by $\mathcal{P}_{i}$ the set of all connected components of
$W_{i}$. We put $\mathcal{C}_{N}=\bigcup_{i\in I_{N}}\mathcal{P}_{i}$. We
notice that connected subsets in $W_{i}$ ($i\in I_{N}$) are also connected in
$X$. It follows that
$\operatorname{ord}(\mathcal{C}_{N})=\operatorname{ord}(\mathcal{W}_{N+m})$
and
$\operatorname{mesh}(\mathcal{C}_{N},d_{B_{r}(\langle\vec{v}\rangle^{\perp})\cap[-N,N]^{k}}^{T})<\epsilon$.
By a similar argument as in Claim 2 (in Lemma 4.4 above) we can find a finite
closed cover $\mathcal{C}^{\prime}_{N}$ of $X$ satisfying that
$\operatorname{ord}(\mathcal{C}^{\prime}_{N})=\operatorname{ord}(\mathcal{C}_{N})$
and
$\operatorname{mesh}(\mathcal{C}^{\prime}_{N},d_{B_{r}(\langle\vec{v}\rangle^{\perp})\cap[-N,N]^{k}}^{T})<\epsilon$.
Clearly, $\mathcal{C}_{N}^{\prime}$ refines $\bigvee_{n\in
B_{r}(\langle\vec{v}\rangle^{\perp})\cap[-N,N]^{k}}T^{n}\alpha$. It follows
from Lemma 2.1 that $\mathcal{D}(\bigvee_{n\in
B_{r}(\langle\vec{v}\rangle^{\perp})\cap[-N,N]^{k}}T^{n}\alpha)\leq\operatorname{ord}(\mathcal{C}^{\prime}_{N})<K\cdot(N+m)^{k-1}$
and thus

$\operatorname{Widim}_{\epsilon^{\prime}}(X,d_{B_{r}(\langle\vec{v}\rangle^{\perp})\cap[-N,N]^{k}}^{T})\leq\operatorname{ord}(\mathcal{C}^{\prime}_{N})<K\cdot(N+m)^{k-1}.$

We remark that the constant $K$ depends only on $(X,\mathbb{Z}^{k},T)$. Since
$m$ is independent of $N$ and since $\epsilon^{\prime}>0$ is arbitrary, the
definition (3.1) allows us to conclude. ∎

To end this paper we remark that we also have the following result which
unifies Meyerovitch and Tsukamoto’s theorem and Kato’s theorem and which does
not include Theorem 1.2. We omit its proof because it is highly similar to
Theorem 1.2 and the previously-known results.

  * •

Let $k$ be a positive integer. If $(X,\mathbb{Z}^{k},T)$ is a continuum-wise
expansive $\mathbb{Z}^{k}$-action and if a $\mathbb{Z}^{k-1}$-action
$(X,\mathbb{Z}^{k-1},R)$ satisfies that $R\colon\mathbb{Z}^{k-1}\times X\to X$
commutes with $T\colon\mathbb{Z}^{k}\times X\to X$ (namely $T^{m}\circ
R^{n}(x)=R^{n}\circ T^{m}(x)$ for all $m\in\mathbb{Z}^{k}$,
$n\in\mathbb{Z}^{k-1}$ and $x\in X$), then $(X,\mathbb{Z}^{k-1},R)$ has finite
mean dimension.

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