# Joint machine learning analysis of muon spectroscopy data from different
materials

T Tula1 G Möller1 J Quintanilla1 S R Giblin2 A D Hillier3 E E McCabe1,4 S
Ramos1 D S Barker1,5 and S Gibson1 1 School of Physical Sciences, University
of Kent, Park Wood Rd, Canterbury CT2 7NH, United Kingdom  2 School of Physics
and Astronomy, Cardiff University, Cardiff CF24 3AA, United Kingdom  3 ISIS
Facility, STFC Rutherford Appleton Laboratory, Chilton, Didcot Oxon, OX11 0QX,
United Kingdom  4 Department of Physics, Durham University, Durham, DH1 3LE,
United Kingdom  5 School of Physics and Astronomy, University of Leeds, Leeds,
LS2 9JT, United Kingdom<EMAIL_ADDRESS>
###### Abstract

Machine learning (ML) methods have proved to be a very successful tool in
physical sciences, especially when applied to experimental data analysis.
Artificial intelligence is particularly good at recognizing patterns in high
dimensional data, where it usually outperforms humans. Here we applied a
simple ML tool called principal component analysis (PCA) to study data from
muon spectroscopy. The measured quantity from this experiment is an asymmetry
function, which holds the information about the average intrinsic magnetic
field of the sample. A change in the asymmetry function might indicate a phase
transition; however, these changes can be very subtle, and existing methods of
analyzing the data require knowledge about the specific physics of the
material. PCA is an unsupervised ML tool, which means that no assumption about
the input data is required, yet we found that it still can be successfully
applied to asymmetry curves, and the indications of phase transitions can be
recovered. The method was applied to a range of magnetic materials with
different underlying physics. We discovered that performing PCA on all those
materials simultaneously can have a positive effect on the clarity of phase
transition indicators and can also improve the detection of the most important
variations of asymmetry functions. For this joint PCA we introduce a simple
way to track the contributions from different materials for a more meaningful
analysis.

##  1 Introduction

In recent years, machine learning tools have become increasingly familiar to
the scientific community, but they are not yet regarded as standard practice
for analysing small-scale experimental measurements in condensed matter
physics. The strength of ML techniques lies in their ability to recognise
patterns and correlations in data [1, 2, 3], which is often the goal of
scientific enquiries (such as the search for phase transitions). Usually, for
the ML methods to work, a large data set is required and since data
acquisition can be costly for many experiments (in both time and resources),
it is understandable that ML tools are not always thought to be a suitable
tool. In this work, we want to test whether experimental muon spectroscopy
measurements can be analysed with ML even when using smaller sets of data.
Specifically we apply a linear principal component analysis algorithm to muon
spectroscopy measurements [4] in order to find signs of phase transitions for
a range of different materials.

In this article we report on a new development from our previous research [5],
in which we demonstrated that PCA can be successfully applied to a range of
different materials simultaneously, thus increasing the effectiveness of this
ML method, even when the underlying physics of the materials are vastly
different. To have more insight into the PCA results in this case, we present
a cumulative principal component score metric, which tells us how important
each PC output is for a given material. That allows us to look at phase
transition indicators in correct principal component scores, and this improved
our previous analysis of magnetic materials.

##  2 Methods

###  2.1 Muon spectroscopy experiments

The main quantity to analyse from muon spectroscopy experiments is the
asymmetry function, which tells us the distribution of spin directions of
muons implanted into the sample. This can be directly linked to the internal
magnetic field of the material studied [6, 7, 8, 9]. We are interested in
studying changes in the asymmetry functions at different temperatures, which
can indicate phase transitions. A more comprehensive introduction to muon
spectroscopy can be found in [4, 5].

###  2.2 PCA

Principal component analysis is an unsupervised machine learning technique [3,
10], which means that we are not required to label the input data, and given
that there are some correlations between asymmetry functions, PCA will be able
to find those. Applying PCA is equal to performing singular value
decomposition on a matrix containing all asymmetry functions $\mathbf{A}$.
Principal components (PC) are given by left-singular vectors of $\mathbf{A}$,
with singular values signifying the importance of each PC. By looking at
projections of asymmetry functions into the most important PCs, we can see
changes in the behaviour that could indicate phase transitions. More on the
PCA method as applied to $\mu$SR data can be found in our previous article [5]

##  3 Results and Discussion

We applied PCA to a range of different materials. For each of them we had
around 10-30 asymmetry functions, measured at different temperatures and at
vanishing external magnetic field throughout. Among the systems studied are
time-reversal symmetry breaking superconductors (LaNiGa2, LaNiC2,
LaNi0.9Cu0.1C2, LaNi0.75Cu0.25C2), a proposed spin liquid (LuCuGaO4) and a
layered antiferromagnet (BaFe2Se2O). We initially tried performing the
analysis for each material individually, but found out that for some materials
the small number of measurements resulted in most PCs having a significant
contribution to covariance. To overcome that, we performed PCA on the joint
dataset of measurements from all materials simultaneously. This improved the
visibility of phase transitions for those measurements that were previously
hard to analyse, and did not significantly alter the other results. This is
remarkable given that the materials in the data set have very different
underlying physics. We can interpret this result as follows: by feeding all
the data to the PCA simultaneously, the algorithm is not learning anything new
about any particular material, but it is learning more about the experimental
technique being employed in all cases ($\mu$SR). A comparison of the shapes of
the ensuing principal components – with PCA performed on one material at a
time or simultaneously on all of them – can be found in our previous work [5].

We now demonstrate that more insight can be gained from the results of a PCA
analysis performed on multiple samples by calculating the total score of a
given PC for each individual material. Unlike analysis for a single material,
where PC scores are naturally ordered in terms of decreasing significance, in
a sample of several materials considered jointly, the order of significance
can depend on the material: Some PC shapes might be dominant in most of
materials, meaning that they will have large contribution to the covariance,
but for other materials another PC might be more prominent. To interpret the
analysis, we should always examine the most important PC(s) for each sample.
To identify the relevant scores, we define a cumulative principal component
score (cPCS) as

$\mbox{cPCS}(i;n)=\sum_{j=1}^{N_{i}}|\mbox{PCS}_{n}^{i}(j)|,$ (1)

where $i$ iterates over different materials, $n$ stands for $n$-th principal
component, PCS${}_{n}^{i}(j)$ corresponds to $n$-th principal component score
for a $j$-th measurement of a given material and $N_{i}$ stands for number of
measurements for $i$-th material. We then use this measure to identify the
principal component scores with the largest cPCS for each material.

Figure 1: a) Temperature dependence for principal component scores when
applying PCA to multiple materials simultaneously. Red vertical lines
correspond to expected phase transitions, taken from the literature; b)
Cumulative principal component scores for all materials.

Our findings are presented in figure 1, which is an extension of figure 5 from
[5]. We present the temperature dependence of the first four principal
component scores for all the materials in a), and their cPCS in b). From the
cPCS we can see that although the 1st PC has the largest contribution to all
of measurements, the 2nd PC is less important than the 3rd one for
Cu0.1LaNi0.9C2 and LuCuGaO4. If we compare the principal component scores for
Cu0.1LaNi0.9C2, we can clearly see that the 3rd PC has a much more visible
indication of the expected phase transition (red vertical line) than the 2nd
or even 1st one. For LuCuGaO4 material, we expected improvement from our
previous joint analysis, and although we find that 3rd PC is more important
than 2nd, there seems to be no feature indication at 0.6 K, which we have
found in principal component analysis of just this material ([5], figure 4,
last column, third row). This tells us that joint analysis is most helpful
whenever the separate PCA is not successful, which we can always tell by
looking at scree plots. In the case of LuCuGaO4, the cPCS metric looks worse
than scree plot of its separate analysis ([5], figure 4, last column, last
row), and we should take that into consideration when analysing the material.
One important note here is that adding LuCuGaO4 did not affect negatively the
results for other materials, which we have showed previously [5].

##  4 Conclusions

We have suggested a new analysis method to extend our recently propsed PCA of
$\mu$SR data [5]. We are more confident that performing a joint PCA of
measurements taken for different materials is a viable method of analysis,
given that for each set of measurements we are looking at the right PC scores,
which can be ensured by calculating a simple metric – cumulative principal
component scores. We should also compare the cPCS to scree plots from separate
analyses conducted for each material, in case the simultaneous PCA might not
improve the clarity of each individual study. The ability to thus enhance our
understanding gained from a new experiment based on existing data goes beyond
the possibilities of preexisting approaches, where data for each material is
necessarily analyzed and fitted in isolation, and overarching commonalities
are anticipated in advance by the formulation of a suitable fitting function.
We anticipate this could offer great advantage when deployed in large-
throughput user facilities.

## References

## References

  * [1] Zdeborová L 2017 Nat Phys 13 420–421
  * [2] Carleo G, Cirac I, Cranmer K, Daudet L, Schuld M, Tishby N, Vogt-Maranto L and Zdeborová L 2019 Rev. Mod. Phys. 91 2773
  * [3] Géron A 2019 Hands-on machine learning with Scikit-Learn, Keras, and TensorFlow: Concepts, tools, and techniques to build intelligent systems (O’Reilly Media)
  * [4] Blundell S J 1999 Muon-spin rotation : a brief introduction [Online; accessed 11\. Sep. 2020] URL http://users.ox.ac.uk/ sjb/musr/musr.html
  * [5] Tula T, Möller G, Quintanilla J, Giblin S R, Hillier A D, McCabe E E, Ramos S, Barker D S and Gibson S 2021 Journal of Physics: Condensed Matter 33 194002
  * [6] Schenck A G E 1985 Muon Spin Rotation Spectroscopy Principles and Applications in Solid State Physics (CRC Press) ISBN 978-085274551-9
  * [7] S F J Cox 1987 Journal of Physics C: Solid State Physics 20 3187
  * [8] Blundell S J 1999 Contemporary Physics 40 175–192
  * [9] Lee S L, Cywinski R and Kilcoyne S H (eds) 1999 Muon science: Muons in physics, chemistry and materials (CRC press)
  * [10] Jolliffe I T and Cadima J 2016 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374 20150202


# Resonant control of polar molecules in an optical lattice

Thomas M. Hanna  Joint Quantum Institute, NIST and University of Maryland, 100
Bureau Drive, Stop 8423, Gaithersburg, MD 20899-8423, USA  Eite Tiesinga
Joint Quantum Institute, NIST and University of Maryland, 100 Bureau Drive,
Stop 8423, Gaithersburg, MD 20899-8423, USA  William F. Mitchell  Applied and
Computational Mathematics Division, National Institute of Standards and
Technology, 100 Bureau Drive Stop 8910, Gaithersburg, Maryland 20899-8910, USA
Paul S. Julienne  Joint Quantum Institute, NIST and University of Maryland,
100 Bureau Drive, Stop 8423, Gaithersburg, MD 20899-8423, USA

###### Abstract

We study the resonant control of two nonreactive polar molecules in an optical
lattice site, focussing on the example of RbCs. Collisional control can be
achieved by tuning bound states of the intermolecular dipolar potential, by
varying the applied electric field or trap frequency. We consider a wide range
of electric fields and trapping geometries, showing that a three-dimensional
optical lattice allows for significantly wider avoided crossings than free
space or quasi-two dimensional geometries. Furthermore, we find that dipolar
confinement induced resonances can be created with reasonable trapping
frequencies and electric fields, and have widths that will enable useful
control in forthcoming experiments.

###### pacs:

03.65.Nk, 34.10.+x, 34.50.Cx

Ultracold gases of polar molecules are of interest for their long-range
dipolar interactions, which give them unique applications in areas such as
many-body phases Baranov (2008), quantum information DeMille (2002), and
precision measurement Sandars (1967); DeMille et al. (2000); Hudson et al.
(2002). Cold gases of LiCs Deiglmayr et al. (2008) and RbCs Sage et al. (2005)
have been formed with temperatures $T\lesssim 1\,$mK, and work continues to
produce degenerate gases Lercher et al. ; Cho et al. . Since the creation of a
near-degenerate gas of 40K87Rb with $T\lesssim 1\,\mu$K K.-K. Ni et al.
(2008), a number of studies have been done of its collision properties K.-K.
Ni et al. (2010); Ospelkaus et al. (2010, 2010); de Miranda et al. (2011). KRb
has an exothermic reaction producing $\mathrm{K}_{2}+\mathrm{Rb}_{2}$, which
occurs with almost unit probability when two molecules are sufficiently close.
This allows a simple description of the collision properties in terms of
universal physics Quéméner and Bohn (2010a, b); Micheli et al. (2010);
Idziaszek et al. (2010); Gao (2010); Kotochigova (2010); Quéméner and Bohn
(2011); Julienne et al. . Such techniques should also apply to other species
with reactive collisions, and to the quenching of any vibrationally excited
molecule. For many studies it is therefore desirable to keep molecules
separated, for example by confining the gas in a three-dimensional (‘3D’)
lattice cho  or in a quasi-two dimensional (‘2D’) geometry with the molecules
polarized perpendicular to the plane (‘side-by-side’) de Miranda et al.
(2011); Quéméner and Bohn (2010a, b); Micheli et al. (2010); Ticknor (2010);
D’Incao and Greene (2011). This reduces the likelihood of molecules
approaching each other along the attractive ‘head-to-tail’ path which is
available in 3D.

In contrast to reactive molecules such as KRb, ground state NaK, NaRb, NaCs,
KCs and RbCs are not reactive, and so are available for experiments on longer
timescales and at higher densities where control of elastic collisions is
useful. The long range dipole-dipole interaction between two molecules
produces an anisotropic potential which is capable of supporting bound states
Kanjilal and Blume (2008). Tuning these bound states around a collision
threshold with an electric field allows resonant control of the interactions
Ticknor and Bohn (2005); Roudnev and Cavagnero (2009); Idziaszek et al.
(2010), in analogy to the magnetic and optical control that has been so useful
for neutral atoms Chin et al. (2010). Because three-body recombination can
still occur Ticknor and Rittenhouse (2010), isolating a pair of molecules in
an optical lattice site provides an ideal, loss-free environment for studying
the two-body energy spectrum. Such a scenario is analogous to several
experiments performed on atom pairs Syassen et al. (2007); Ospelkaus et al.
(2006); Thalhammer et al. (2006); Volz et al. (2006). Optical lattices have
also been used to tune atomic collisions through confinement induced
resonances Olshanii (1998); Petrov et al. (2000); sal ; Haller et al. (2010);
Fröhlich et al. (2011) (CIRs), which depend on the scattering length being
comparable to the characteristic length of the confinement. With the
interactions of polar molecules having an even longer range, it is reasonable
to anticipate easy creation of a CIR.

In this paper we study the states and control possibilities of two polar
molecules isolated in an optical lattice site, focussing on the specific
example of RbCs. We examine in detail the effects of tuning the lattice
parameters and electric field. We show that the optical lattice can be used to
increase the resonance width past what is possible in free space or 2D
geometries. We compare the eigenenergies obtained for a quasi-2D lattice site
to scattering calculations for a system with confinement in only one
direction, which accurately reproduce the avoided crossings and show their
utility for resonant control. Our studies show that tuning the confinement has
a significant effect on the collisional and bound state properties of the pair
of molecules, allowing the creation of useful CIRs.

Quantity | Definition | Value ($a_{0}$)
---|---|---
Mean scattering length | $\bar{a}=\frac{2\pi}{[\Gamma(1/4)]^{2}}(2m_{\textrm{r}}C_{6}/\hbar^{2})^{1/4}$ | 233.5
Confinement length | $\ell_{\mathrm{ho}}=\sqrt{\hbar/(2m_{\textrm{r}}\omega)}$ |
$\omega/2\pi=1$ kHz |  | 5728
$\omega/2\pi=50$ kHz |  | 810.5
Dipole length | $a_{\mathrm{\mu}}=m_{\textrm{r}}\mu^{2}/\hbar^{2}$ | $3.1\times 10^{4}$
Table 1: Characteristic length scales for the interaction of ultracold
molecules in an optical lattice, with values given for RbCs in parameter
regimes used in the present work. For the van der Waals coefficient, we use
$C_{6}=142129E_{h}a_{0}^{6}$ Kotochigova (2010), where $E_{h}=4.3597\times
10^{-18}$ J is the Hartree energy and $a_{0}=52.918$ pm is the Bohr radius. We
give the confinement length for optical lattice sites with frequencies
$\omega/2\pi=1\,$kHz and 50 kHz. The dipole length is given for RbCs molecules
with a dipole moment of $\mu=1.0\,$D, where
$\textrm{D}=0.39343ea_{0}=3.336\times 10^{-30}$ Cm is the Debye and $e$ is the
charge of an electron. Here, $m_{\textrm{r}}$ is the reduced mass.

We consider two ground state 87Rb133Cs molecules in a cylindrically symmetric
optical lattice site. In Table 1 we list length scales relevant to ultracold
molecular collisions and give representative values for the parameter regimes
used in this work. Typical van der Waals coefficients for polar molecules are
of order $10^{5}\,E_{h}a_{0}^{6}$ – $10^{7}\,E_{h}a_{0}^{6}$, much larger than
those for pairs of alkali atoms ($10^{3}\,E_{h}a_{0}^{6}$ –
$10^{4}\,E_{h}a_{0}^{6}$). However, the mean scattering length scales as
$\bar{a}\propto C_{6}^{1/4}$, giving a similar characteristic length to the
van der Waals part of the potential. We take the dipole moment
$\mu=\langle\hat{\mu}\rangle_{\textrm{z}}$ to be the expectation value of the
electric dipole operator $\hat{\mu}$ for the molecular ground state in the
electric field direction. We calculate this electric-field dependent quantity
according to the method of Ref. boh . The dipole length, tunable with an
electric field, is typically the largest length scale in the problem. For a
trapping frequency $\omega/2\pi=50\,$kHz, $a_{\mu}=\ell_{\textrm{ho}}$ for
$\mu=0.16\,$D. We note that the use of a strong dipole moment takes us beyond
the region of validity of pseudopotential approaches such as those of Refs.
Kanjilal et al. (2007); Derevianko (2003).

The molecules are assumed to be rigid rotors, aligned in the axial direction
by an applied electric field. We approximate the lattice site with a harmonic
trap and consider only the relative motion of the molecules. The combined
interaction and trapping potential is given by

$\displaystyle V(\rho,z)$
$\displaystyle=\frac{\mu^{2}}{r^{3}}\left(1-\frac{3z^{2}}{r^{2}}\right)+\frac{C_{12}}{r^{12}}-\frac{C_{6}}{r^{6}}+\frac{\hbar^{2}}{2m_{\textrm{r}}}\frac{m^{2}}{\rho^{2}}$
$\displaystyle+\tfrac{1}{2}m_{\textrm{r}}(\omega_{\rho}^{2}\rho^{2}+\omega_{z}^{2}z^{2})\,.$
(1)

Here, $\rho$ and $z$ are the relative radial and axial coordinates,
respectively. Also, $\omega_{\rho,\textrm{z}}=2\pi f_{\rho,\mathrm{z}}$, where
$f_{\rho,\textrm{z}}$ are the corresponding trapping frequencies. The
intermolecular separation is given by $r=\sqrt{\rho^{2}+z^{2}}$, and the
projection of the relative motion along the axis of symmetry is given by $m$.
We impose a repulsive short-range potential, $C_{12}/r^{12}$, setting the
$C_{12}$ coefficient such that the potential $C_{12}/r^{12}-C_{6}/r^{6}$
contains six bound states and gives a scattering length of 100$a_{0}$. We
neglect the anisotropic $C_{6}$ coefficient. While arbitrary, setting the
short range part of the potential in this way allows us to conveniently study
the important long range effects.

Although the collisions under consideration involve four atoms, the approach
described above is justified by the separation in energy scale between the
chemical bonds within the ground state dimers ($\sim$THz) and the collision
energy or bond between them ($\lesssim\,$MHz). We also note that the van der
Waals coefficient between polar molecules has contributions from the rotation
of the molecules as well as the induced dipole moments of the electron clouds.
An electric field polarizes the molecules and changes the rotational
contribution. We have calculated the extent of this change and checked that it
does not noticeably change the results presented here, as was the case in Ref.
Julienne et al. . Consequently, we neglect this effect. Because we confine
ourselves to a single collision channel, the resonances we find correspond to
shape resonances Chin et al. (2010), in which the potential experienced by a
colliding pair supports a near-degenerate quasi-bound state. We note that
Feshbach resonances, in which a colliding pair is coupled to a near-degenerate
bound state of a different spin configuration, are possible for the general
case of coupling between states of different molecular spin and rotational
quantum number.

We first study the two-body energy spectrum. We solve for eigenstates and
eigenvalues of the Hamiltonian with the potential of Eq. (1) using PHAML
Version 1.8.0 pha ; Mitchell and Tiesinga (2005), a parallel two-dimensional
finite element code for elliptic boundary value and eigenvalue problems. PHAML
features adaptive grid refinement of the discretized spatial coordinates to
concentrate the grid in areas where the wave function varies rapidly, and high
order elements to obtain an accurate solution. For these computations we used
eighth degree elements. Within PHAML, ARPACK leh  was used to solve the
discrete eigenvalue problem, using the shift-and-invert spectral
transformation to compute interior eigenvalues, and MUMPS Amestoy et al.
(2001) to solve the resulting linear system of equations. The parallel
computations were performed on two nodes of a Linux cluster. A particular
advantage of the two-dimensional solver is its ability to readily account for
the anisotropic interaction and trapping potential. By contrast, an expansion
in spherical harmonics or non-interacting trap states will struggle to
accurately resolve the wavefunction without a very large basis set. However,
for analysis of the wavefunctions we calculate projections onto these
functions. We solve for the function $F(\rho,z)$, where the full wavefunction
is given by $\psi(\rho,z,\phi)=F(\rho,z)e^{im\phi}$. Our bound state
calculations consider only $m=0$, but in the scattering calculations described
below we will consider the effects of nonzero $m$. We study spherically
symmetric ($\omega_{z}=\omega_{\rho}$) and quasi-2D
($\omega_{z}\gg\omega_{\rho}$) geometries, with the dipoles always aligned
along the $z-$axis.

Figure 1: (a) Eigenenergies for two RbCs molecules in an optical lattice site
with $f_{z}=f_{\rho}=25\,$kHz. Trap states are adiabatically converted to
bound states as the dipolar interaction is increased. Points marked ‘b’ and
‘c’ correspond to the wavefunctions shown in the lower panels, which
illustrate the head-to-tail configuration at two different dipole moments. For
the three lowest energy trap states, the partial wave at $\mu=0$ is indicated.
We also give the partial waves into which these states have a significant
projection after being converted to bound states as $\mu$ is increased. Red
crosses indicate the non-interacting trap state energies. In (b) and (c), we
plot $\rho|\psi|^{2}$, and scale all lengths by the confinement length in the
$z$-direction, $\ell_{\textrm{ho}}^{\textrm{z}}$.

The eigenenergies of two RbCs molecules in a spherically symmetric lattice
site with $f_{z}=f_{\rho}=25\,$kHz are shown as a function of dipole moment in
Fig. 1a. We use the term bound states to refer to those that are bound when
the trap is adiabatically turned off. States close to the non-interacting trap
level energies,
$(n_{\textrm{z}}+1/2)\hbar\omega_{\textrm{z}}+(2n_{\rho}+|m|+1)\hbar\omega_{\rho}$,
are called trap states. Here, Bose symmetry allows the trap state quantum
numbers to be $n_{\textrm{z}}=0,2,4\ldots$, $n_{\rho}=0,1,2,\ldots$ and
$m=0,\pm 2,\pm 4,\ldots$. At zero dipole moment, the trap state energies are
affected by the van der Waals interactions. Dipole moments above approximately
0.1 D cause a significant change to these energies. A large number of avoided
crossings occur as trap states are brought into the potential by the
increasing dipolar attraction. All crossings are avoided, although some are
too narrow to be visible on the scale shown. States of different $\ell$ are
mixed by the dipolar interactions, with the broadest crossings occuring
between states of low $\ell$. For example, the first two trap levels converted
to bound states as $\mu$ is increased are primarily of mixed $s$\- and
$d$-wave symmetry, whereas the steeply descending state with a series of
narrow crossings near $\mu=0.31\,$D is concentrated in $\ell=10$. Figures 1b
and 1c illustrate the wavefunction near $E/h=-50\,$kHz for dipole moments of
$\mu=0.17\,$D and $\mu=0.498\,$D, respectively. We plot the function
$\rho|\psi|^{2}$, to more clearly illustrate both short and large length
scales. For the state at $\mu=0.17\,$D, vibrational nodes of the bound states
of the van der Waals potential can be seen as rings of constant
$r<0.25\ell_{\textrm{ho}}^{\textrm{z}}$. A $d$-wave component at long range
provides the head-to-tail configuration, with the wavefunction concentrated in
the region $\rho<|z|$. This makes $E$ decrease as $\mu$ increases. The state
at $\mu=0.498\,$D is much more strongly coupled between partial waves, and has
a correspondingly more complicated configuration.

Figure 2: Elastic collision rate as a function of $\mu$ for a trapping
frequency of $f_{\textrm{z}}=50$ kHz and collision energy of
$k_{\textrm{B}}\times 200$ nK, showing the contributions of the $m=0$, $\pm 2$
and $\pm 4$ partial waves (upper panel), and the corresponding $m=0$ bound
state calculation with $f_{z}=50\,$kHz and $f_{\rho}=1\,$kHz (lower panel).
The $z$ trap states with a substantial admixture are indicated for each bound
state not concentrated solely in $n_{\textrm{z}}=0$.

We now compare these results to the case of a quasi-2D optical lattice site,
with $f_{\textrm{z}}=50\,$kHz and $f_{\rho}=1\,$kHz. Eigenvalues are plotted
as a function of $\mu$ in the lower panel of Fig. 2. A quasi-continuum of
radial trap levels is found, spaced by $2f_{\rho}$, instead of the strongly
mixed states of Fig. 1. Only the $n_{\textrm{z}}=0$ trap state is within the
energy range shown, although some bound states have substantial admixture in
higher $z$ states, as indicated in the lower panel of Fig. 2. The quasi-2D
configuration has the effect of making the avoided crossings narrower than in
the 3D lattice case. An intuitive explanation of this effect is that a
sufficiently strong dipole moment allows molecules to overcome the confinement
which holds them side-by-side, and move to the attractive head-to-tail
configuration. The overlap between the asymptotic states representing these
two configurations is reduced by a higher trap aspect ratio.

It is desirable to attach meaningful quantum numbers to describe the states
that are observed. This is made difficult at large $\mu$ by the strong
anisotropic interactions; however, some approximate quantum numbers can be
used. States can be described by their projections onto the non-interacting
trap levels with quantum numbers $n_{\textrm{z}}$ and $n_{\rho}$, although
both van der Waals and dipole-dipole interactions mix these levels. We
indicate in Fig. 2 the states which are concentrated in higher $z$ trap
levels. As we discuss below, it is these states that provide the possibility
of creating CIRs. States can also be described by their vibrational quantum
number and projections onto spherical harmonics described by the orbital
angular momentum $\ell$, which are good quantum numbers in the zero-dipole
limit. Different $\ell$ also become strongly mixed at sufficient $\mu$, as
discussed above for the states shown in Fig. 1.

We now make the link between our calculated bound state energies and
scattering properties in a quasi-2D system. Our scattering calculations use
the coupled channels technique discussed in Julienne et al. , adapted for
elastic boundary conditions at short range. We use the potential of Eq. (1)
with $\omega_{\rho}$ set to 0, and propagate the scattering matrix in a basis
of spherical harmonics to a value of $r$ large enough to match onto the
$n_{\textrm{z}}=0$ trap state. The chosen collision energy of
$k_{\textrm{B}}\times 200$ nK, corresponding to $h\times 4$ kHz, is such that
higher $z$ trap states are not significantly populated at this separation.
Here, $k_{\textrm{B}}$ is the Boltzmann constant. We then propagate outwards
in $\rho$, extracting the scattering properties at long range using the
conventional tools of scattering theory Taylor (1972). The results are shown
in the upper panel of Fig. 2, and agree well with the bound state
calculations. The resonances at low dipole moment are widest and most isolated
from other features, making them the most useful for resonant control.
Contributions to the elastic collision rate coefficient from collisions with
higher $m$ make the resonance minima nonzero.

Figure 3: Confinement induced resonances created by tuning of
$f_{\textrm{z}}$. The top panel shows the elastic collision rate for a
quasi-2D trap with $f_{\textrm{z}}=50$ kHz. We have summed the contributions
of partial waves from $m=0$ to $|m|=4$. Solid lines correspond to
$\mu=0.3048\,$D and collision energies of $k_{\textrm{B}}\times 1\,$nK,
$k_{\textrm{B}}\times 100\,$nK and $k_{\textrm{B}}\times 200\,$nK, as
labelled. The dashed line ($\mu=0.3058\,$D) shows the sensitivity of the
resonance location to electric field variation. The bottom panel shows the
eigenenergies of the Schrödinger equation with the potential of Eq. (1), using
$\mu=0.3048$ D and trapping frequencies of $f_{\mathrm{z}}=50$ kHz and
$f_{\rho}=1$ kHz. The dashed green line shows the perturbative result of Eq.
(2). Dotted red lines and arrows show the intersections of this calculation
with scattering states of the given kinetic energies. These intersections
agree well with the calculated resonance locations of the upper panel.

Optical lattices have been used to control scattering lengths in neutral gases
Haller et al. (2010); Fröhlich et al. (2011) by making a state corresponding
to an excited trap level near degenerate with the colliding atoms. This
depends on the scattering length being of the same order as the characteristic
length scale of the confinement. With the large dipole length characteristic
of interactions between polar molecules, it is reasonable to expect that
similar confinement induced effects should occur. In the lower panel of Fig. 3
we show this effect for RbCs molecules, calculating eigenenergies as a
function of $f_{\textrm{z}}$ while maintaining $f_{\rho}=1$ kHz. We assume
$\mu=0.3048$ D, corresponding to an easily accessible electric field of 0.67
kV/cm. The figure shows six trap levels with $n_{\textrm{z}}=0$ and a range of
$n_{\rho}$, and a single bound state crossing these levels. As shown in Fig.
2, this bound state has substantial admixture in higher $z$ trap states. The
energy of the bound state therefore increases with $f_{\textrm{z}}$ faster
than the trap levels.

We have perturbatively calculated the change $\delta E$ in an eigenenergy from
changing $\omega_{\textrm{z}}$ to
$\omega_{\textrm{z}}+\delta\omega_{\textrm{z}}$, which results in

$\displaystyle\delta
E=\frac{1}{2}\left\langle\left(\frac{z}{\ell_{\textrm{ho}}^{\textrm{z}}}\right)^{2}\right\rangle\hbar\delta\omega_{\textrm{z}}\,.$
(2)

Here $\langle\cdots\rangle$ indicates calculating the expectation value with
respect to the selected wavefunction. The result for the bound state of Fig.
3, evaluated with the numerically obtained wavefunction at
$f_{\textrm{z}}=40\,$kHz, is shown as a green dashed line. The red dotted
lines correspond to the energy of a non-interacting pair of molecules in the
ground trap state, with $k_{\textrm{B}}\times 1$ nK and $k_{\textrm{B}}\times
200$ nK of relative kinetic energy. The points at which these cross the
perturbative calculation correspond well with the scattering resonances shown
in the upper panel of Fig. 3. For a relative kinetic energy of
$k_{\textrm{B}}\times 200$ nK, the feature has a width with respect to
$f_{\textrm{z}}$ variation of approximately 5 kHz, although accurate electric
field control will be necessary due to the resonance location being strongly
dependent on the dipole moment. This is shown by the dashed line in the top
panel, for a dipole moment of 0.3058 D, which corresponds to a change in the
applied electric field of approximately 2.5 V/cm. The temperature is also of
significance, as shown by the calculations for collision energies of
$k_{\textrm{B}}\times 100$ nK and $k_{\textrm{B}}\times 1$ nK, which produce
narrower peaks at lower trapping frequencies. This result illustrates that the
location and properties of the resonances can be controlled by manipulating
both the electric field and the confinement.

In conclusion, we have studied the role that an optical lattice can play in
controlling the collisional properties of nonreactive polar molecules. We have
shown that tight confinement allows for much broader avoided crossings, giving
a greater resonance width than is available in free space. We have also shown
that confinement induced resonances can be easily created, with the caveat
that their location is sensitive to the dipole moment. Measurements of
resonance locations would constrain the short range potential, for which we
studied just one example with a scattering length of 100 $a_{0}$. However,
this should not significantly alter the density of states or our finding that
RbCs will have several accessible resonances for dipole moments less than 0.5
D and trapping frequencies on the order of tens of kHz. These results will
therefore be of significance for upcoming experiments using non-reactive polar
molecules.

We acknowledge funding from an AFOSR MURI on ultracold molecules (T.M.H. and
P.S.J.) and partial funding from the ONR (P.S.J.). We thank Z. Idziaszek for
stimulating discussions.

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# Weighted Mean Topological Dimension 00footnotetext: 2010 Mathematics Subject
Classification: 37B40, 37C45

Yunping Wang
School of Mathematical Sciences and Institute of Mathematics Nanjing Normal
University
Nanjing 210046 Jiangsu  P.R.China
e-mail<EMAIL_ADDRESS>
###### Abstract

This paper is devoted to the investigation of the weighted mean topological
dimension in dynamical systems. We show that the weighted mean dimension is
not larger than the weighted metric mean dimension, which generalizes the
classical result of Lindenstrauss and Weiss [16]. We also establish the
relationship between the weighted mean dimension and the weighted topological
entropy of dynamical systems, which indicates that each system with finite
weighted topological entropy or small boundary property has zero weighted mean
dimension.

Keywords: weighted mean topological dimension; weighted entropy; small
boundary property

##  1 Introduction

One of the basic invariants of a dynamical system $(X,T)$ is its topological
entropy. This quantifies to what extent nearby points diverge as the system
evolves. Adler, Konheim and McAndrew [1] first introduced it as an invariant
of topological conjugacy for studying dynamical systems in compact topological
spaces. Subsequently, Dinaburg [7] and Bowen [4] presented a different, yet
equivalent, definition in metric spaces. Kenyon and Peres [14] studied the
dimension of invariant sets and measures on the torus under expanding maps. In
[3, 8], Barral and Feng defined weighted topological pressure via relative
thermodynamic formalism and subadditive thermodynamic formalism, in the
particular case when the underlying dynamical systems $(X,T)$ and $(Y,S)$ are
subshifts over finite alphabets. Motivated by this work, Feng and Huang [9]
extended to general topological systems and introduced a weighted version of
entropy, which provided a topological extension of dimension theory of
invariant sets and measures on the torus under affine diagonal endomorphisms.
For convenience, we recall the definition of the weighted entropy here.

Let $k\geq 2$. Assume that $(X_{i},d_{i}),i=1,\cdots,k$, are compact metric
spaces, and $(X_{i},T_{i})$ are topological dynamical systems. Moreover,
assume that for each $1\leq i\leq k-1,(X_{i+1},T_{i+1})$ is a factor of
$(X_{i},T_{i})$ with a factor map $\pi_{i}:X_{i}\to X_{i+1};$ in other words,
$\pi_{1},\cdots,\pi_{k-1}$ are continuous maps such that the following
diagrams commute,

$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{T_{1}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\scriptstyle{T_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{k-1}}$$\textstyle{X_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T_{k}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{k-1}}$$\textstyle{X_{k}}$

For convenience, we use $\pi_{0}$ to denote the identity map on $X_{1}$.
Define $\tau_{i}:X_{1}\to X_{i+1}$ by
$\tau_{i}=\pi_{i}\circ\pi_{i-1}\circ\cdots\circ\pi_{0}$ for
$i=0,1,\cdots,k-1.$ Let $M(X_{i},T_{i})$ denote the set of all
$T_{i}$-invariant Borel probability measures on $X_{i}$ and $E(X_{i},T_{i})$
denote the set of ergodic measures. Fix
$\bf{a}$$=(a_{1},a_{2},\cdots,a_{k})\in\mathbb{R}^{k}$ with $a_{1}>0$ and
$a_{i}\geq 0$ for $i\geq 2.$ For $\mu\in M(X_{1},T_{1})$, we call

$h^{\bf{a}}_{\mu}(T_{1}):=\sum_{i=1}^{k}a_{i}h_{\mu\circ\tau_{i-1}^{-1}}(T_{i})$

the $\bf{a}$-weighted measure-theoretic entropy of $\mu$ with respect to
$T_{1}$, or simply, the $\bf{a}$-weighted entropy of $\mu$, where
$h_{\mu\circ\tau_{i-1}^{-1}}(T_{i})$ denotes the measure-theoretic entropy of
$\mu\circ\tau_{i-1}^{-1}$ with respect to $T_{i}.$

######  Definition 1.1.

[9] ($\bf{a}$-weighted Bowen ball). For $x\in
X_{1},n\in\mathbb{N},\epsilon>0,$ let

$\displaystyle B_{n}^{\bf{a}}(x,\epsilon):=\\{y\in
X_{1}:d_{i}(T_{i}^{j}\tau_{i-1}x,T_{i}^{j}\tau_{i-1}y)<\epsilon\leavevmode\nobreak\
\leavevmode\nobreak\ \text{for}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle 1\leq i\leq k\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle 0\leq
j\leq\lceil(a_{1}+\cdots+a_{i})n\rceil-1\\},$

where $\lceil u\rceil$ denotes the least integer $\geq u$. We call
$B_{n}^{\bf{a}}(x,\epsilon)$ the $n$-th $\bf{a}$-weighted Bowen ball of radius
$\epsilon$ centered at $x$.

Return back to the metric spaces $(X_{i},d_{i})$ and topological dynamical
systems $(X_{i},T_{i}),i=1,2,\cdots,k.$ For $n\in\mathbb{N}$, define a metric
$d_{n}^{\bf a}$ on $X_{1}$ by

$\displaystyle d_{n}^{\bf
a}(x,y)=\sup\\{d_{i}(T_{i}^{j}\tau_{i-1}x,T_{i}^{j}\tau_{i-1}y):1\leq i\leq
k,0\leq j\leq\lceil(a_{1}+\cdots+a_{i})n\rceil-1\\}.$

######  Definition 1.2.

Let $n$ be a natural number, $\epsilon>0$,
$\bf{a}$$=(a_{1},a_{2},\cdots,a_{k})\in\mathbb{R}^{k}$ and $K$ be a compact
subset of $X_{1}$. A subset $F$ of $X_{1}$ is said to $({\bf a},n,\epsilon)$
spanning set of $K$ with respect to $T_{1}$ if for all $x\in K$, there exists
$y\in F$ with $d_{n}^{\bf a}(x,y)\leq\epsilon$. Let $r_{n}^{\bf
a}(\epsilon,K)$ denote the smallest cardinality of any $({\bf a},n,\epsilon)$
spanning set for $K$ with respect to $T_{1}$.

######  Definition 1.3.

Let $n$ be a natural number, $\epsilon>0$,
$\bf{a}$$=(a_{1},a_{2},\cdots,a_{k})\in\mathbb{R}^{k}$and $K$ be a compact
subset of $X_{1}$. A subset E of $X_{1}$ is said to be $({\bf a},n,\epsilon)$
separated with respect to $T_{1}$ if $x,y\in E,x\neq y$, implies $d_{n}^{\bf
a}(x,y)>\epsilon$. Let $s_{n}^{\bf a}(\epsilon,K)$ denote the largest
cardinality of any $({\bf a},n,\epsilon)$ separated set for $K$ with respect
to $T_{1}$.

The ${\bf a}$-weighted topological entropy $h^{\bf a}(X_{1})$ using spanning
sets and separated sets is defined as follows.

$\displaystyle h^{\bf a}(X_{1})$
$\displaystyle=\lim\limits_{\epsilon\rightarrow
0}\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\log{r_{n}^{\bf
a}}(X_{1},\epsilon)$ $\displaystyle=\lim\limits_{\epsilon\rightarrow
0}\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\log{s_{n}^{\bf
a}}(X_{1},\epsilon).$

Mean dimension is a conjugacy invariant of topological dynamical systems which
was introduced by Gromov [10]. In 2000, Lindenstrauss and Weiss [16] used it
to answer an open question raised by Auslander [2] that whether every minimal
system $(X,T)$ can be imbedded in $[0,1]^{\mathbb{Z}}$. It turns out that mean
dimension is the right invariant to study for the problem of existence of an
embedding into $(([0,1]^{D})^{\mathbb{Z}},\sigma)$. More applications and
interesting relations have been found [11, 17, 12, 13]. The metric mean
dimension was introduced in [16] and they proved that metric mean dimension is
an upper bound of the mean dimension. It allowed them to establish the
relationship between the mean dimension and the topological entropy of
dynamical systems, which shows that each system with finite topological
entropy has zero mean dimension. This invariant enables one to distinguish
systems with infinite topological entropy. In [15, 16], they also introduced
the small boundary property (SBP) by replacing the empty set with the small
set in the definition of zero dimensional space, which could be seen as a
dynamical version of being totally disconnected, and indicated that a
dynamical system with this property has zero mean dimension. In order to
investigate the embedding question of weighted topological dynamical systems,
we will introduce the concepts of the weighted mean dimension and weighted
metric mean dimension inspired by the ideas of [9, 16].

In this paper, we introduce the mean dimension for weighted version which
include the classic case as $a_{1}=1,a_{i}=0$, $i\geq 2$. We also define the
weighted metric mean dimension and show that topological dynamical system
$(X_{1},T_{1})$ with finite weighted topological entropy, or small boundary
property has zero weighted mean dimension. Meanwhile, we obtain an inequality
between the weighted mean dimension and the weighted metric mean dimension.
The following theorems present the main results of this paper.

######  Theorem 1.4.

For any dynamical system $(X_{1},T_{1})$ and any metric $d_{i}$ compatible
with the topology on $X_{i},\ i=1,2,\cdots,k,$

${\rm mdim}^{\bf a}(X_{1},T_{1})\leq{\rm mdim}_{M}^{\bf a}(X_{1},d),$

where the definitions of these notions will be given in Section 3.

######  Theorem 1.5.

If $(X_{1},T_{1})$ has the SBP, the weighted mean dimension is zero.

The remainder of this paper is organized as follows. In Section 2, we recall
definitions and properties of the mean dimension. In Section 3, we define the
weighted mean dimension and weighted metric mean dimension, investigate the
relationship between the two notions, and show that finite weighted
topological entropy implies the zero weighted mean dimension. In Section 4, we
introduce the small boundary property, and devote to proving that the SBP
yields the zero wighted mean dimension. In Section 5, we present an example.

##  2 Preliminaries

Next we recall some basic facts about mean dimension. These notions and
terminologies can be found in [6] and [16]. Let $X$ be a compact metric space
and $\alpha$ be a finite open cover of $X$, we call that a cover $\beta$
refines $\alpha$$(\beta\succ\alpha)$, if every member of $\beta$ is a subset
of some member of $\alpha$.

######  Definition 2.1.

If $\alpha$ is an open cover of $X$, we shall denote

$ord(\alpha)=\max\limits_{x\in X}\sum\limits_{U\in\alpha}1_{U}(x)\ \text{and}\
D(\alpha)=\min\limits_{\beta\succ\alpha}ord(\beta),$

where $\beta$ runs over all finite open covers of $X$ refining $\alpha$.

######  Definition 2.2.

Let $X$ be a topological space. The topological dimension $dim(X)$ of $X$ is
the quantity defined by

$dim(X)=\sup\limits_{\alpha}D(\alpha)$,

where $\alpha$ runs over all finite covers of $X$.

######  Definition 2.3.

If $(X,T)$ is a dynamical system, then the mean dimension of $(X,T)$, denoted
by $\text{mdim}(X,T)$, is defined by

${\rm
mdim}(X,T)=\sup\limits_{\alpha}\lim\limits_{n\rightarrow\infty}\frac{1}{n}D(\bigvee\limits_{i=0}^{n-1}T^{-i}\alpha),$

where $\alpha$ runs over all finite open covers of $X$.

######  Definition 2.4.

Let $X$ and $Y$ be topological spaces. Let $\alpha$ be a finite open cover of
$X$. A continuous map $f:X\rightarrow Y$ will be called $\alpha$-compatible if
it is possible to find a finite open cover $\beta$ of $f(X)$ such that
$f^{-1}(\beta)\succ\alpha.$ We will use the notation $f\succ\alpha$ to denote
that $f$ is $\alpha$-compatible.

######  Proposition 2.5.

Let $X$ and $Y$ be topological spaces and let $f:X\rightarrow Y$ be a
continuous map. Let $\beta$ be a finite open cover of $Y$. Then one has

$D(f^{-1}(\beta))\leq D(\beta)$.

######  Proposition 2.6.

Let $\alpha$ and $\beta$ be finite open covers of $X$, then

$D(\alpha\vee\beta)\leq D(\alpha)+D(\beta)$.

######  Proposition 2.7.

If $X$ is compact, $f:X\rightarrow Y$ a continuous function such that for
every $y\in Y$, $f^{-1}(y)$ is a subset of some $U\in\alpha$, then $f$ is
$\alpha$-compatible.

######  Proposition 2.8.

If $\alpha$ is an open cover of $X$, then

$D(\alpha)\leq k$

iff there is an $\alpha$-compatible continuous function $f:X\rightarrow K$
where $K$ has topological dimension $k$.

##  3 Proof of Theorem 1.4

Firstly, we introduce the concepts of weighted mean dimension and weighted
metric mean dimension. Let $k\geq 2$. Assume that
$(X_{i},d_{i}),i=1,2,\cdots,k$, are compact metric spaces, and $(X_{i},T_{i})$
are topological dynamical systems. Moreover, assume that for each $1\leq i\leq
k-1,(X_{i+1},T_{i+1})$ is a factor of $(X_{i},T_{i})$ with a factor map
$\pi_{i}:X_{i}\to X_{i+1}$. Fix ${\bf
a}=(a_{1},\cdots,a_{k})\in\mathbb{R}^{k}$ with $a_{1}>0$ and $a_{i}\geq 0$ for
$2\leq i\leq k$. Let $\alpha_{i}$ be a finite open cover of $(X_{i},T_{i})$
for $1\leq i\leq k$. We define the weighted mean dimension as follows.

######  Definition 3.1.

${\rm mdim}^{\bf
a}(X_{1},T_{1})=\sup\limits_{\alpha_{1},\cdot\cdot\cdot,\alpha_{k}}\limsup\limits_{n\rightarrow\infty}\dfrac{1}{n}{D(\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})n\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i})}$,
where $\alpha_{i}$ is a finite open cover of $X_{i},1\leq i\leq k$.

######  Remark 3.2.

When $a_{1}=1,a_{i}=0$ $(i\geq 2)$, the weighted mean dimension recovers the
classic mean dimension. Our interest is on the general case that $a_{i}\neq 0$
$(1\leq i\leq k)$. It is worth pointing out that weighted mean dimension is
different from mean dimension in the case of $a_{i}\neq 0$ $(2\leq i\leq k)$.
In fact, we know the sequence
$\\{D(\bigvee\limits_{i=0}^{n-1}T^{-i}\alpha)\\}_{n\geq 1}$ is subadditive,
however, the sequence

$\\{D(\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})n\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i})\\}_{n\geq
1}$

is not subadditive, then the limit in the definition 3.1 doesn’t exist in case
of $a_{i}\neq 0$ for some $2\leq i\leq k.$

######  Proposition 3.3.

Let $X_{1}$ be a compact metric space and $T_{1}:X_{1}\rightarrow X_{1}$ a
continuous map. Then one has ${\rm mdim}^{\bf
a}(X_{1},T_{1}^{n})=n\cdot\text{\rm mdim}^{\bf a}(X_{1},T_{1})$ for every
positive integer $n$.

###### Proof.

For $1\leq i\leq k$, let $\alpha_{i}$ be a finite open cover of $X_{i}$. Set

$\displaystyle
w(\\{{\alpha_{i}}\\}_{i=1}^{k},T_{1},n)=\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})n\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}.$

Since $\lceil(a_{1}+\cdots+a_{i})mn\rceil-1\geq
n(\lceil(a_{1}+\cdots+a_{i})m\rceil-1)$ for every integer $m\geq 1$, we have
the open cover

$\displaystyle
w(\\{{\alpha_{i}}\\}_{i=1}^{k},T_{1},mn)=\bigvee_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})(mn)\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}$

is finer than the open cover

$\displaystyle
w(\\{{\alpha_{i}}\\}_{i=1}^{k},T_{1}^{n},m)=\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})m\rceil-1}(T_{1}^{n})^{-j}\tau_{i-1}^{-1}\alpha_{i}.$

It follows that

$\displaystyle
D(\bigvee_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})(mn)\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i})\geq
D(\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})m\rceil-1}(T_{1}^{n})^{-j}\tau_{i-1}^{-1}\alpha_{i}).$

We deduce that

$\displaystyle n\cdot\text{mdim}^{\bf a}(X_{1},T_{1})\geq\text{mdim}^{\bf
a}(X_{1},T_{1}^{n}).$ (1)

On the other hand, by
$\lceil(a_{1}+a_{2}+....+a_{i})nm\rceil-1\leq\lceil(a_{1}+a_{2}+....+a_{i})m\rceil
n-1$, we obtain

$\displaystyle\bigvee_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})nm\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}\prec\bigvee_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})m\rceil
n-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}$

for each $1\leq i\leq k$. Let
$\beta_{i}=\bigvee\limits_{j=0}^{n-1}T_{i}^{-j}\alpha_{i}$, then

$\displaystyle w(\\{{\alpha_{i}}\\}_{i=1}^{k},T_{1},mn)\prec
w(\\{{\beta_{i}}\\}_{i=1}^{k},(T_{1})^{n},m).$

Moreover,

$\displaystyle
n\dfrac{1}{mn}D(\bigvee_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})(mn)\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i})\leq\dfrac{1}{m}D(\bigvee_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})m\rceil-1}(T_{1}^{n})^{-j}\beta_{i}).$

Then

$\displaystyle n\cdot\text{mdim}^{\bf a}(X_{1},T_{1})\leq\text{mdim}^{\bf
a}(X_{1},T_{1}^{n}).$ (2)

Inequalities (1) and (2) imply that

$\displaystyle n\cdot\text{mdim}^{\bf a}(X_{1},T_{1})=\text{mdim}^{\bf
a}(X_{1},T_{1}^{n}).$

∎

######  Proposition 3.4.

Let $\text{dim}(X_{1})<\infty$. Then $\text{\rm mdim}^{\bf a}(X_{1},T_{1})=0$.

###### Proof.

For every finite open cover $\alpha_{i}$ of $X_{i}$, $i=1,...,k$, we have

$\displaystyle
D(\bigvee_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})n\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i})\leq\text{dim}(X_{1}).$

As $\text{dim}(X_{1})<\infty$, we deduce that

$\displaystyle\limsup\limits_{n\rightarrow\infty}\dfrac{1}{n}D(\bigvee_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+....+a_{i})n\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i})=0.$

Thus, we have that

$\displaystyle\text{mdim}^{\bf a}(X_{1},T_{1})=0.$

∎

We define a weighted version of metric mean dimension. For an open cover
$\alpha$, define the mesh of $\alpha$ according to the metric $d_{n}^{\bf a}$
by

$mesh(\alpha,d_{n}^{\bf a})=\max\limits_{U\in\alpha}diam(U)$.

Set

$\displaystyle S^{{\bf
a}}(X_{1},\epsilon)=\limsup\limits_{n\rightarrow\infty}\inf\limits_{mesh(\alpha,d_{n}^{\bf
a})<\epsilon}\frac{1}{n}\log|\alpha|.$ (3)

It is obvious that $S^{{\bf a}}(X_{1},\epsilon)$ is monotone nondecreasing as
$\epsilon\rightarrow 0$.

######  Definition 3.5.

We define the weighted metric mean dimension of $X_{1}$,

$\displaystyle{\rm mdim}_{M}^{\bf
a}(X_{1},d)=\liminf\limits_{\epsilon\rightarrow 0}\frac{S^{{\bf
a}}(X_{1},\epsilon)}{|\log\epsilon|}.$ (4)

Recall how the weighted topological entropy is defined for a dynamical system
$X_{1}$. According to Definition 1.2, set

$r_{n}^{\bf a}(X_{1},\epsilon)=\min\\{{|F|:F\subset X_{1}\ \text{is}\ ({\bf
a},n,\epsilon)\text{-spanning set}}\\},$ $r^{\bf
a}(X_{1},\epsilon)=\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\log
r_{n}^{\bf a}(X_{1},\epsilon),$

and

$h^{\bf a}(X_{1})=\lim\limits_{\epsilon\rightarrow 0}r^{\bf
a}(X_{1},\epsilon)$ .

Notice that

$r^{\bf a}(X_{1},\epsilon)\geq S^{{\bf a}}(X_{1},2\epsilon)\geq r^{\bf
a}(X_{1},2\epsilon)$.

Thus (4) is equivalent to

$\text{mdim}^{\bf a}_{M}(X_{1},d)=\liminf\limits_{\epsilon\rightarrow
0}\frac{r^{\bf a}(X_{1},\epsilon)}{|\log\epsilon|}$.

######  Proposition 3.6.

If ${\rm mdim}^{\bf a}_{M}(X_{1},d)\neq 0$, then $h^{\bf a}(X_{1})=\infty.$

######  Corollary 3.7.

If $X_{1}$ has finite weighted topological entropy, then ${\rm mdim}^{\bf
a}_{M}(X_{1},d)=0$.

The next result is inspired in the Theorem 4.2 of [16] and shows that the
weighted metric mean dimension is an upper bound for the weighted mean
dimension.

###### Proof of Theorem 1.4.

Assume that for each $1\leq i\leq k-1$, $(X_{i+1},T_{i+1})$ is a factor of
$(X_{i},T_{i})$ with a factor map $\pi_{i}:X_{i}\rightarrow X_{i+1}$. Define
$\tau_{i}:X_{1}\rightarrow X_{i+1}$ by
$\tau_{i}=\pi_{i}\circ\pi_{i-1}\circ\cdots\circ\pi_{0}$ for
$i=0,1,\cdots,k-1$. As

$\text{mdim}^{\bf
a}(X_{1},T_{1})=\sup\limits_{\alpha_{1},\cdots,\alpha_{k}}\limsup\limits_{n\to\infty}\dfrac{1}{n}D(\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})n\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}),$

it is sufficient to show that for any finite open cover $\alpha_{i}$ of
$X_{i},\ i=1,2,\cdots,k$,

$\limsup\limits_{n\to\infty}\dfrac{1}{n}D(\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})n\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i})\leq\text{mdim}_{M}^{\bf
a}(X_{1},d).$

Since $\alpha_{i}$ be a finite open cover of $X_{i}$, then
$\tau_{i-1}^{-1}\alpha_{i}$ is a finite open cover of $X_{1}$. We can refine
$\tau_{i-1}^{-1}\alpha_{i}$ to be of the form

$\tau_{i-1}^{-1}\alpha_{i}=\\{U_{i_{1}},V_{i_{1}}\\}\vee\\{U_{i_{2}},V_{i_{2}}\\}\vee\cdots\vee\\{U_{i_{r_{i}}},V_{i_{r_{i}}}\\},$

where for $1\leq j\leq r_{i}$, $1\leq i\leq k$, $\\{U_{i_{j}},V_{i_{j}}\\}$ is
an open cover of $X_{1}$ . Set $w_{i,j}:\ X_{1}\rightarrow[0,1]$ by

$\displaystyle w_{i,j}(x)=\dfrac{d(x,X_{1}\backslash
V_{i_{j}})}{d(x,X_{1}\backslash V_{i_{j}})+d(x,X_{1}\backslash U_{i_{j}})},\
1\leq i\leq k,\leavevmode\nobreak\ 1\leq j\leq r_{i}.$ (5)

Then $w_{i,j}$ is Lipschitz, and

$\displaystyle\begin{split}U_{i_{j}}&=w_{i,j}^{-1}[0,1)\\\
V_{i_{j}}&=w_{i,j}^{-1}(0,1].\end{split}$ (6)

Let $C_{L}$ be a bound on the Lipschitz constants of all $w_{i,j}$.

For any $N$ define $F(N,\cdot):\
X_{1}\rightarrow[0,1]^{\sum\limits_{i=1}^{k}r_{i}\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}$
by

$\displaystyle F(N,x)=($ $\displaystyle w_{1,1}(x),\cdots,w_{1,r_{1}}(x),$
$\displaystyle w_{1,1}(T_{1}x),\cdots,w_{1,r_{1}}(T_{1}x),\cdots$
$\displaystyle w_{1,1}(T_{1}^{\lceil
a_{1}N\rceil-1}x),\cdots,w_{1,r_{1}}(T_{1}^{\lceil a_{1}N\rceil-1}x)$
$\displaystyle\cdots$ $\displaystyle w_{k,1}(x),\cdots,w_{k,r_{k}}(x),$
$\displaystyle w_{k,1}(T_{1}x),\cdots,w_{k,r_{k}}(T_{1}x),$ $\displaystyle
w_{k,1}(T_{1}^{\lceil(a_{1}+a_{2}+\cdots+a_{k})N\rceil-1}x),\cdots,w_{k,r_{k}}(T_{1}^{\lceil(a_{1}+a_{2}+\cdots+a_{k})N\rceil-1}x)).$

From Proposition 2.7 and (6) we see that
$F(N,\cdot)\succ\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}$.
Usually, if

$S\subset\\{1,\cdots,\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}\\},$

then $F(N,x)_{S}\in[0,1]^{|S|}$ is the projection of $F(N,x)$ to the
coordinates in the index set $S$.

######  Claim 3.8.

Let $\epsilon>0$, $\ D={\rm mdim}_{M}^{\bf a}(X_{1},d)$. If $N$ is larger than
some $N(\epsilon)$, there exists
$\xi\in(0,1)^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}}$
such that for any $|S|\geq(D+\epsilon)N$,

$\xi_{S}\notin F(N,X)_{S}.$

###### Proof of Claim 3.8.

Let

$\delta<(2^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})\rceil}}(2C_{L})^{2D})^{-\frac{2}{\epsilon}},$

such that

$\dfrac{S(X_{1},\epsilon,d_{n}^{\bf a})}{|\log\delta|}\leq\text{mdim}_{M}^{\bf
a}(X_{1},d)+\epsilon/4.$

Fix $N$ large enough (depending on $\epsilon$ and $\delta$), $X_{1}$ can be
covered by $\delta^{-(D+\epsilon/2)N}$ weighted Bowen balls of the form

$\displaystyle B_{N}^{\bf a}(x,\delta)=$ $\displaystyle\\{x^{\prime}\in
X_{1}:d_{i}(T_{i}^{j}\tau_{i-1}x,T_{i}^{j}\tau_{i-1}x^{\prime})<\delta,\ \
\forall 1\leq i\leq k\ \text{and}\ $ $\displaystyle 0\leq
j\leq\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1\\}.$

Notice that $C_{L}$ is a Lipschitz constant for all $w_{i,j}$, we have

$F(N,B_{N}^{\bf
a}(x,\delta))\subset\\{a\in[0,1]^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}}:\|F(N,x)-a\|_{\infty}\leq
C_{L}\delta\\}.$

Then the set $F(N,X_{1})$ can be covered by $\delta^{-N(D+\epsilon/2)}$ balls
in the $\parallel\cdot\parallel_{\infty}$ norm of radius $C_{L}\delta.$
Enumerate these weighted Bowen balls as $B^{\bf a}(k),\
k=1,\cdots,K=\delta^{-(D+\epsilon/2)N}.$ Choose $\xi$ with uniform probability
in
$[0,1]^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}}$.

Then for a set
$S\subset\\{1,\cdots,\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}\\},$

$P(\xi_{S}\in F(N,X_{1})_{S})\leq\sum\limits_{k=1}^{K}P(\xi_{S}\in B^{\bf
a}(k)|_{S})\leq\delta^{-(D+\epsilon/2)N}(2C_{L}\delta)^{|S|}.$

Hence

$\displaystyle P($ $\displaystyle\exists S:|S|\geq(D+\epsilon)N\ \text{and}\
\xi_{S}\in F(N,X_{1})_{S})$
$\displaystyle\leq\sum\limits_{|S|\geq(D+\epsilon)N}P(\xi_{S}\in
F(N,X_{1})_{S})$ $\displaystyle\leq(\sharp\ \text{of such
}S)\times\delta^{-(D+\epsilon/2)N}(2C_{L}\delta)^{(D+\epsilon)N}$
$\displaystyle\leq
2^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}}\times\delta^{-(D+\epsilon/2)N}(2C_{L}\delta)^{(D+\epsilon)N}$
$\displaystyle\leq
2^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}}\times((2C_{L})^{2D}\delta^{\epsilon/2})^{N}\ll
1,$

and so, with high probability, a random $\xi$ will satisfy the requirements. ∎

######  Claim 3.9.

If
$\pi:F(N,X_{1})\rightarrow[0,1]^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}}$
satisfies

$\displaystyle\\{1\leq
k\leq\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}:\xi_{k}=a\\}\subset$
$\displaystyle\\{1\leq
k\leq\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}:\pi(\xi)|_{k}=a\\},$

for both $a=0$ and $1$, and all
$\xi\in[0,1]^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}}.$
Then $\pi\circ F(N,X_{1})$ is compatible with
$\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}$.

###### Proof of Claim 3.9.

In fact, if
$\xi\in[0,1]^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}},$
define for $0\leq m_{i}<\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil$ and $1\leq
n_{i}<r_{i}$

$W_{m_{i},n_{i}}=\begin{cases}T_{1}^{-m_{i}}U_{n_{i}}&\text{if
}\xi_{m_{i}r_{i}+n_{i}}=0,\\\ T_{1}^{-m_{i}}V_{n_{i}}&\text{otherwise.}\\\
\end{cases}$

It is easy to see that

$F^{-1}\pi^{-1}(\xi)\subset\bigcap\limits_{i=1}^{k}\bigcap\limits_{0\leq
m_{i}<\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil,\atop 1\leq
n_{i}<r_{i}}W_{m_{i},n_{i}}\in\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}.$

∎

Let $\epsilon>0$. Choose $\overline{\xi},N$ as in Claim 3.8. Set

$\Phi=\\{\xi\in[0,1]^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}}:\xi_{k}=\overline{\xi}_{k}\
\text{for more than }(D+\epsilon)N\ \text{indexes }k\\};$

thus
$F(N,X_{1})\subset[0,1]^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}}\backslash\Phi$.
We denote the latter set briefly by $\Phi^{C}$. We shall construct a
continuous retraction $\pi$ of $\Phi^{c}$ onto the
$\lfloor(D+\epsilon)N\rfloor+1$ skeleton of the cube
$I=[0,1]^{\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}}$.
For $1\leq
m\leq\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}$,
define

$J_{m}=\\{\xi\in I:\xi_{i}\in\\{0,1\\}\ \text{for at least }m\ \text{indexes
}1\leq
i\leq\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}\\}.$

Since $\overline{\xi}$ is in the interior of $I$, one can define
$\pi_{1}:I\backslash\\{\overline{\xi}\\}\rightarrow J_{1}$ by mapping each
$\xi$ to the intersection of the ray starting at $\overline{\xi}$ and passing
through $\xi$ and $J_{1}$. For each of the
$(\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}-1)$-dimensional
cubes $I^{l}$ that comprise $J_{1}$, we can define a retraction on $I^{l}$ in
a similar fashion using as a center the projection of $\overline{\xi}$ onto
$I^{l}$. This will define a continuous retraction $\pi_{2}$ of $\Phi^{C}$ into
$J_{2}$. As long as there is some intersection of $\Phi$ with the cubes in
$J_{m}$ this process can be continued, thus we get finally a continuous
projection $\pi$ of $\Phi^{C}$ onto $J_{m_{0}}$, with

$m_{0}+\lfloor(D+\epsilon)N\rfloor+1=\sum\limits_{i=1}^{k}r_{i}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}.$

Obviously, $\pi$ satisfies the conditions of Claim 3.9. Therefore

$\pi\circ
F(N,\cdot)\succ\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}.$

Moreover, as $F(N,\cdot)\subset\Phi^{C}$,

$\pi\circ F(N,\cdot)\subset J_{m_{0}},$

the latter having topological dimension $\lfloor(D+\epsilon)N\rfloor+1$.

In conclusion, we have constructed a
$\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}$
compatible function from $X_{1}$ to a space of topological dimension
$\leq$$(D+\epsilon)N+1$, and according to Proposition 2.8,

$D(\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i})\leq(D+\epsilon)N+1.$

As $\epsilon>0$ can be chosen as small as we like, $\alpha_{i}$ as defined as
we like for $1\leq i\leq k$, we conclude that

$\text{mdim}^{\bf a}(X_{1},T_{1})\leq D.$

∎

##  4 Proof of Theorem 1.5

In this section, we discuss the relationship between the small-boundary
property and weighted mean dimension.

######  Definition 4.1.

Let $(X,T)$ be a dynamical system, and $E$ a subset of $X$. We define the
orbit capacity of the set $E$ to be

$ocap(E)=\lim\limits_{n\to\infty}\sup\limits_{x\in
X}\dfrac{1}{n}\sum_{i=0}^{n-1}1_{E}(T^{i}x).$

A set $E$ will be called small if $ocap(E)=0$.

######  Definition 4.2.

A dynamical system $(X,T)$ has the small boundary property (SBP) if every
point $x\in X$ and every open set $U\ni x$ there is a neighborhood $V\subset
U$ of $x$ with small boundary.

######  Proposition 4.3.

[16] If $(X,T)$ has the SBP, then for every open cover $\alpha$ of $X$ and
every $\epsilon$ there is a subordinate partition of unity $\phi_{j}:\
X\rightarrow[0,1],\ (j=1,\cdots,|\alpha|)$ such that

$\displaystyle\sum_{j=1}^{|\alpha|}\phi_{j}(x)=1,$ $\displaystyle
supp(\phi_{j})\subset U\ \text{for some}\ U\in\alpha,\ j=1,\cdots,|\alpha|,$
(7) $\displaystyle
ocap(\bigcup\limits_{j=1}^{|\alpha|}\phi_{j}^{-1}(0,1))<\epsilon.$

###### Proof of Theorem 1.5.

Assume that for each $1\leq i\leq k-1$, $(X_{i+1},T_{i+1})$ is a factor of
$(X_{i},T_{i})$ with a factor map $\pi_{i}:X_{i}\rightarrow X_{i+1}$. Define
$\tau_{i}:X_{1}\rightarrow X_{i+1}$ by
$\tau_{i}=\pi_{i}\circ\pi_{i-1}\circ\cdots\circ\pi_{0}$ for
$i=0,1,\cdots,k-1$. Let $(X_{1},T_{1})$ have the SBP and $\alpha_{i}$ is a
finite open cover of $X_{i}$ for $1\leq i\leq k$. Given $\epsilon>0$.
According to the Proposition 4.3, we can construct a partition of unity
subordinate to $\beta_{i}=\tau_{i-1}^{-1}\alpha_{i}$ such that satisfies (7)
for $i=1,\cdots,k$. Let $|\beta_{i}|=c_{i}$ and
$A=\bigcup\limits_{m=1}^{c_{i}}\phi_{m,i}^{-1}(0,1)$ for each $1\leq i\leq k$.
Let $N$ is large enough such that

$\displaystyle\dfrac{1}{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}\sum\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}1_{A}(T_{1}^{j}(x))<\epsilon,\
\forall x\in X_{1},i=1,\cdots,k.$ (8)

Let $\Phi_{i}:(X_{1},T_{1})\rightarrow\mathbb{R}^{c_{i}}$ is given by

$x\rightarrow(\phi_{1,i}(x),\cdots,\phi_{c_{i},i}(x)).$

Given $1\leq i\leq k$, define the map
$f_{N}^{i}:X_{1}\rightarrow\mathbb{R}^{c_{i}\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}$
by

$f_{N}^{i}(x)=(\Phi(x),\Phi(T_{1}x),\cdots,\Phi(T_{1}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}x)).$

We claim that $f_{N}^{i}(X_{1})$ is a subset of a finite number of $\epsilon
c_{i}\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil$ dimensional affine subspaces of
$\mathbb{R}^{c_{i}\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}$.

Indeed, let $e_{l}^{m},\ l=1,\cdots,\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil,\
m=1,\cdots,c_{i}$ be the standard base of
$\mathbb{R}^{c_{i}\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}$. Define for every
$I=\\{i_{1},i_{2},\cdots,i_{N^{\prime}}\\},\
N^{\prime}<\epsilon\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil$, and every
$\xi\in\\{0,1\\}^{c_{i}\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil}$

$C(I,\xi,i)=\text{span}\\{e_{l}^{m}:\ l\in I,\ 1\leq m\leq c_{i}\\}+\xi.$

Then by (8),

$f_{N}^{i}(X_{1})\subset\bigcup\limits_{|I|<\epsilon\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil,\
\xi}C(I,\xi,i).$

It is obvious that $f_{N}^{i}$ is
$\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}T_{1}^{-j}\beta_{i}$-compatible.
According to Proposition 2.8, we have

$D(\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}T_{1}^{-j}\beta_{i})<\epsilon
c_{i}\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil.$

By subadditive of $D(\alpha)$, we thus get

$\displaystyle
D(\bigvee\limits_{i=1}^{k}\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}T_{1}^{-j}\beta_{i})$
$\displaystyle\leq\sum\limits_{i=1}^{k}D(\bigvee\limits_{j=0}^{\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil-1}T_{1}^{-j}\beta_{i})$
$\displaystyle\leq\sum\limits_{i=1}^{k}\epsilon
c_{i}\lceil(a_{1}+a_{2}+\cdots+a_{i})N\rceil.$

Since $\sum\limits_{i=1}^{k}c_{i}<\infty$, it follows that $(X_{1},T_{1})$ has
zero weighted mean dimension. ∎

##  5 Example

######  Example 5.1.

Put ${\bf a}=\left(1,1\right)$. Assume that $X_{1}=([0,1]^{2})^{\mathbb{Z}}$
and $X_{2}=[0,1]^{\mathbb{Z}}$. Define $T_{1}:X_{1}\rightarrow X_{1}$ by
$T_{1}((x_{m})_{m\in\mathbb{Z}})=(x_{m+1})_{m\in\mathbb{Z}},$ and define
$T_{2}:X_{2}\rightarrow X_{2}$ by
$T_{2}((a_{m})_{m\in\mathbb{Z}})=(a_{m+1})_{m\in\mathbb{Z}}$. Let $\tau_{1}$
be the canonical projection from $X_{1}$ to $X_{2}$, that is to say,
$\tau_{1}((x_{m}^{1},x_{m}^{2})_{m\in\mathbb{Z}})=(x_{m}^{1})_{m\in\mathbb{Z}}$.
Then $(X_{2},T_{2})$ is the factor of $(X_{1},T_{1})$ associated with the
factor map $\tau_{1}$. We define a distance $d_{1}$ on $X_{1}$ by

$d_{1}(x,y)=\sum\limits_{m\in\mathbb{Z}}2^{-|m|}d^{\prime}(x_{m},y_{m}),\ \ \
\ (x=(x_{m})_{m\in\mathbb{Z}},y=(y_{m})_{m\in\mathbb{Z}}\in X_{1}),$

where $d^{{}^{\prime}}(s,t)=\max\\{|s_{1}-t_{1}|,|s_{2}-t_{2}|\\}$ is a
distance on $[0,1]^{2}$. We define a distance $d_{2}$ on $X_{2}$ by

$d_{2}(a,b)=\sum\limits_{m\in\mathbb{Z}}2^{-|m|}|a_{m}-b_{m}|,\ \ \ \
(a=(a_{m})_{m\in\mathbb{Z}},b=(b_{m})_{m\in\mathbb{Z}}\in X_{2}).$

We calculate the weighted metric mean dimension of $X_{1}$. Let $\epsilon>0$
and set $l=\lfloor\log_{2}(\frac{4}{\epsilon})\rfloor$. Then
$\sum\limits_{|n|\leq l}2^{-|n|}\leq\frac{\epsilon}{2}$. We consider an open
covering of $[0,1]^{2}$ by

$D_{\frac{\epsilon}{6}}(k,k^{{}^{\prime}})=(\frac{(k-1)\epsilon}{12},\frac{(k+1)\epsilon}{12})\times(\frac{(k^{{}^{\prime}}-1)\epsilon}{12},\frac{(k^{{}^{\prime}}+1)\epsilon}{12}),\leavevmode\nobreak\
\leavevmode\nobreak\ 1\leq
k,k^{{}^{\prime}}\leq\lfloor\frac{12}{\epsilon}\rfloor.$

For $n\geq 1$, consider

$([0,1]^{2})^{\mathbb{Z}}=\bigcup\limits_{1\leq
k_{-l},\cdots,k_{2n+l}\leq\frac{12}{\epsilon}\atop 1\leq
k^{{}^{\prime}}_{-l},\cdots,k^{{}^{\prime}}_{2n+l}\leq\frac{12}{\epsilon}}\\{x|x_{-l}\in
D_{\frac{\epsilon}{6}}(k_{-l},k^{{}^{\prime}}_{-l}),\cdots,x_{2n+l}\in
D_{\frac{\epsilon}{6}}(k_{2n+l},k^{{}^{\prime}}_{2n+l})\\}.$

We define $\sharp(X,d,\epsilon)$ as the minimum cardinality $N$ of the open
covering $\left\\{U_{1},\cdots,U_{N}\right\\}$ of $X$ such that all $U_{n}$
have diameter smaller than $\epsilon$. Each open set in the right hand side
has diameter less than $\epsilon$ with respect to the distance $d_{n}^{\bf
a}$. Hence

$\sharp([0,1]^{\mathbb{Z}},d_{n}^{\bf
a},\epsilon)\leq(1+\lfloor\frac{12}{\epsilon}\rfloor)^{4n+4l+2}=(1+\lfloor\frac{12}{\epsilon}\rfloor)^{4n+4\lfloor\log_{2}(\frac{4}{\epsilon})\rfloor+2}.$

On the other hand, any two distinct points in the sets

$\\{x\in([0,1]^{2})^{\mathbb{Z}}|x_{m}\in(k_{1},k_{2})\ \text{for}\
\text{all}\ 0\leq m<2n\\}$

where
$k_{i}\in\\{0,\epsilon,2\epsilon,\cdots,\lfloor\frac{1}{\epsilon}\rfloor\epsilon\\},$
have distance $\geq\epsilon$ with respect to $d_{n}^{\bf a}.$ It follows that
$\sharp([0,1]^{\mathbb{Z}},\epsilon)\geq(1+\lfloor\frac{1}{\epsilon}\rfloor)^{4n}.$
Therefore

$S(X_{1},\epsilon)=\limsup\limits_{n\rightarrow\infty}\frac{1}{n}\log\sharp(X_{1},d_{n}^{\bf
a},\epsilon)\sim 4|\log\epsilon|\ \ \ (\epsilon\rightarrow 0).$

Thus ${\rm mdim}_{M}^{\bf a}(X_{1},d)={\rm mdim}_{M}^{\bf(1,1)}(X_{1},d)=4$.

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