Multiplicity of topological systems

David Burguet Sorbonne Universite, LPSM, 75005 Paris, France david.burguet@upmc.fr and Ruxi Shi Sorbonne Universite, LPSM, 75005 Paris, France ruxi.shi@upmc.fr

Abstract.

We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number $k$ of real continuous functions $f_{1},\cdots,f_{k}$ such that the functions $f_{i}\circ T^{n}$ , $n\in\mathbb{Z}$ , $1\leq i\leq k,$ span a dense linear vector space in the space of real continuous functions on $X$ endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.

2020 Mathematics Subject Classification:

1. Introduction

The multiplicity of an invertible bounded operator $U:E\circlearrowleft$ on a normed vector space $E$ is the minimal cardinality of subsets $F\subset E$ , whose cyclic space (i.e. the vector space spanned by $U^{k}x$ , $k\in\mathbb{Z}$ , $x\in F$ ) is dense in $E$ .

For an ergodic measure preserving system $(X,f,\mathcal{B},\mu)$ , the multiplicity $\text{\rm Mult}(\mu)$ of the Koopman operator, which is the operator of composition by $f$ on the Hilbert space $L^{2}(\mu)$ is a dynamical invariant, which has been investigated in many works (see e.g. [Dan13] and the references therein).

Cyclicity, which corresponds to simple multiplicity (i.e. there is an element whose cyclic space is dense in the whole vector space), has been also established for operators of composition on the Hardy space $H^{2}(D)$ [BS97]. In this context a pioneering work of Birkhoff [Bir29] states that there is an entire function $\phi$ in the complex plane such that the set $\{\phi(\cdot+n),\ n\in\mathbb{N}\}$ is dense itself in the set of entire functions endowed with the uniform topology on compact subsets, i.e. the operator of translation by $1$ is hypercyclic.

Quite surprisingly the corresponding topological invariant has not been studied in full generality. More precisely we consider here topological dynamical systems $(X,T)$ , where $X$ is a compact metrizable space and $T:X\circlearrowleft$ is a homeomorphism and we study the operator of composition by $T$ on the Banach space $C(X)$ of real continuous functions endowed with the uniform topology. We call topological multiplicity of $(X,T)$ the associated multiplicity and we denote it by $\text{\rm Mult}(T)$ . We remark that our definitions and results can be extended to the noninvertible continuous map $T:X\circlearrowleft$ . But for sake of simplicity, we focus on homeomorphisms $T:X\circlearrowleft$ .

In this paper, we mostly focus on topological systems with finite multiplicity. We first show the following properties for such systems.

Theorem.

Let $(X,T)$ be a topological system with finite multiplicity. Then the following properties are satisfied.

(1)

$(X,T)$ has zero topological entropy.
(2)

$(X,T)$ has finitely many ergodic measures.

These properties are the main contents of Section 2. The property (1) is proven in Proposition 3.5 in two ways: one uses the variational principal of topological entropy; the other is purely topological. The property (2) is proven in Lemma 2.6 and Corollary 2.8. In fact we show more precisely that the number of ergodic measures is equal to the multiplicity of the operator induced on the quotient of $C(X)$ by the closure of coboundaries.

In Section 3, we relate the topological multiplicity with the dimension of cubical shifts, in which the action $T_{*}:\mathcal{M}(X)\circlearrowleft$ induced by $T$ on the set $\mathcal{M}(X)$ of Borel probability measures on $X$ may be affinely embedded. In Theorem 3.3, we show a necessary and sufficient condition for the existence of affinely embedding of $(\mathcal{M}(X),T_{*})$ to the shift on $([0,1]^{d})^{\mathbb{Z}}$ . Furthermore, we compare our result to Lindenstrauss-Tsukamoto conjecture for dynamical embedding (Corollary 3.4).

In Section 4, we state a generalized Banach version of a lemma due to Baxter [Bax71] which is a classical criterion of simplicity for ergodic transformations. The generalized Baxter’s Lemma (Lemma 4.1) will play an important role on estimating the topological multiplicity in next sections.

For minimal Cantor systems a topological analogue of the rank of a measure preserving system has been defined and studied (see [DP22]). In Section 5, under this setting, we compare the topological multiplicity with the topological rank (Theorem 5.1).

In Section 6, we study some examples and estimate their topological multiplicity: minimal rotations on compact groups, Sturmian and substitution subshifts, homeomorphisms of the interval, etc.

In Section 7, we estimate the topological multiplicity of subshifts with linear growth complexity, i.e. subshifts $X$ such that the cardinality $p_{X}(n)$ of $n$ -words in $X$ satisfies $\liminf_{n\to\infty}\frac{p_{X}(n)}{n}<+\infty$ . Such subshifts aroused a great deal of interest, specially recently [Bos92, CK19, CP23, DDMP21]. In [Bos92] it is proved that an aperiodic subshift $X$ has at most $k$ ergodic measures if $\liminf_{n}\frac{p_{X}(n)}{n}\leq k\in\mathbb{N}$ . Our main related result states as follows (Theorem 7.1 and Theorem 7.6):

Theorem.

Let $X$ be an aperiodic subshift with $\liminf_{n\to\infty}\frac{p_{X}(n)}{n}\leq k\in\mathbb{N}$ . Then

\text{\rm Mult}(T)\leq 2k\text{ and }\sum_{\mu\ ergodic}\text{\rm Mult}(\mu)\leq 2k.

Except the results on multiplicity that we investigate, we propose several questions in the current paper.

2. Topological multiplicity, definition and first properties

2.1. Multiplicity of a linear operator

Let $(E,\|\cdot\|)$ be a normed vector space over $\mathbb{R}$ . We consider a linear invertible bounded operator $U:E\circlearrowleft$ . A subset $F$ of $E$ is called a generating family of $U$ when the vector space spanned by $U^{k}x$ , $k\in\mathbb{Z}$ , $x\in F$ , is dense in $E$ . In the following we denote by $\text{span}(G)$ (resp. $\overline{\text{span}}(G)$ ) the vector space spanned by a subset $G$ of $E$ (resp. its closure) and we then let $V^{U}_{F}:=\overline{\text{span}}\{U^{k}x:k\in\mathbb{Z},x\in F\}$ . Sometimes, we write $V_{F}$ instead of $V^{U}_{F}$ whenever the operator is fixed. The multiplicity $\text{\rm Mult}(U)\in\mathbb{N}\cup\{\infty\}$ of $U$ is then the smallest cardinality of generating families of $U$ . By convention we let $\text{\rm Mult}(U)=0$ when $E$ is reduced to $\{0\}$ . A linear operator with multiplicity one is called cyclic.

We first study the equivariant map between two normed vector spaces with linear invertible bounded operators.

Lemma 2.1.

Let $U_{i}:E_{i}\circlearrowleft$ , $i=1,2$ be two linear invertible bounded operators. Assume that there is a linear bounded operator $W:E_{1}\rightarrow E_{2}$ satisfying $W\circ U_{1}=U_{2}\circ W$ then

\text{\rm Mult}(U_{2}|_{\overline{\mathrm{Im}(W)}})\leq\text{\rm Mult}(U_{1}).

Proof.

One checks easily that if $F$ is a generating family for $U_{1}$ then $W(F)$ is a generating family for the restriction of $U_{2}$ to the closure of the image of $W$ . Therefore $\text{\rm Mult}(U_{2}|_{\overline{\mathrm{Im}(W)}})\leq\text{\rm Mult}(U_{1})$ . ∎

A direct consequence of Lemma 2.1 is that the multiplicity is a spectral invariant : if $U_{i}$ are linear invertible operators on $E_{i}$ $,i=1,2$ , satisfying $W\circ U_{1}=U_{2}\circ W$ for some invertible bounded linear operator $W:E_{1}\rightarrow E_{2}$ , then $U_{1}$ and $U_{2}$ have the same multiplicities.

When $E^{\prime}$ is a closed subspace of $E$ we endow the quotient $E/E^{\prime}$ space with the norm $\|\overline{u}\|^{\prime}=\inf\{\|u+v\|,\ v\in E^{\prime}\}$ . If $E^{\prime}$ is invariant by $U$ we let $U^{E/E^{\prime}}$ be the action induced by $U$ on the quotient normed space $E/E^{\prime}$ . In this context, by applying Lemma 2.1 with $W:E\rightarrow E/E^{\prime}$ being the natural projection, we get

\cdot

\displaystyle\text{\rm Mult}(U^{E/E^{\prime}})\leq\text{\rm Mult}(U).

2.2. Operator of composition : Topological and ergodic multiplicities.

Ergodic theory focuses on the study of invertible measure preserving systems $(X,f,\mathcal{B},\mu)$ . In particular the spectral properties of the unitary operator $U_{f}:L^{2}(\mu)\circlearrowleft$ , $\phi\mapsto\phi\circ f$ , are investigated. We let $\|f\|_{2}:=(\int_{X}|f(x)|^{2}d\mu)^{1/2}$ be the $L^{2}$ -norm of $f\in L^{2}(\mu)$ .

Definition 2.2.

The ergodic multiplicity $\text{\rm Mult}(\mu)$ of an ergodic system $(X,f,\mathcal{B},\mu)$ is the multiplicity of the restriction of $U_{f}$ to the Hilbert space $L_{0}^{2}(\mu):=\{f\in L^{2}(\mu),\ \int f\,d\mu=0\}$ , that is to say, $\text{\rm Mult}(\mu)=\text{\rm Mult}(U_{f})$ .

This quantity has been intensely studied in ergodic theory (see Danilenko’s survey [Dan13]).

Next we consider here an invertible topological dynamical system $(X,T)$ , i.e. $T:X\circlearrowleft$ is a homeomorphism of a compact metric space $X$ . We denote by $C(X)$ the Banach space of real continuous functions endowed with the topology of uniform convergence. We let $\|f\|_{\infty}:=\sup_{x\in X}|f(x)|$ be the supremum norm of $f\in C(X)$ .

Definition 2.3.

The topological multiplicity $\text{\rm Mult}(T)$ of $(X,T)$ is the multiplicity of the operator of composition $U_{T}:C(X)\circlearrowleft$ , $\phi\mapsto\phi\circ T$ .

Quite surprisingly this last notion seems to be new (note however that cyclicity of $U_{T}$ has already been investigated in some cases). Let us first observe that the topological multiplicity bounds from above the ergodic multiplicity of ergodic $T$ -invariant measures.

Lemma 2.4.

Let $(X,T)$ be an invertible topological dynamical system. For any ergodic $T$ -invariant measure $\mu$ , we have

\text{\rm Mult}(\mu)\leq\text{\rm Mult}(T).

Proof.

Let $F$ be a generating family with minimal cardinality of $U_{T}:C(X)\circlearrowleft$ . Then the vector space spanned by $F$ is dense in $(C(X),\|\cdot\|_{\infty})$ , therefore in $(L^{2}(\mu),\|\cdot\|_{2})$ . As $p:L^{2}(\mu)\rightarrow L^{2}_{0}(\mu)$ , $f\mapsto f-\int f\ d\mu$ is continuous and $p\circ U_{T}=U_{T}\circ p$ , the vector space spanned by $p(F)$ is dense in $L^{2}_{0}(\mu)$ .

∎

Let $\mathcal{M}(X)$ be the set of Borel probability measures endowed with the weak- $*$ topology. It is standard that $\mathcal{M}(X)$ is a compact metrizable space. The compact subset $\mathcal{M}(X,T)\subset\mathcal{M}(X)$ of Borel $T$ -invariant probability measures of $(X,T)$ is a simplex, whose extreme set is given by the subset $\mathcal{M}_{e}(X,T)$ of ergodic measures. A topological system with a unique (ergodic) invariant measure is said to be uniquely ergodic. Jewett-Krieger theorem states that every ergodic system has a uniquely ergodic model. Several proofs have been given of this theorem, e.g. see [DGS06, Section 29]. One may wonder if the multiplicity may be preserved:

Question 2.5.

Given an ergodic system with measure $\mu$ , is there a uniquely ergodic model $(X,T)$ of it such that $\text{\rm Mult}(T)=\text{\rm Mult}(\mu)$ ?

2.3. The number of ergodic measures as a multiplicity.

Let $(X,T)$ be an invertible topological dynamical system. A function $\psi\in C(X)$ is a called a continuous $T$ -coboundary, if $\psi$ is equal to $\phi\circ T-\phi$ for some $\phi\in C(X)$ . In other terms the set $B_{T}(X)$ of continuous $T$ -coboundaries is the image of $U_{T}-\text{\rm Id}$ , in particular it is a vector space. Observe that $U_{T}(B_{T}(X))=B_{T}(X)$ . To simplify the notations we write $\tilde{U}_{T}$ for the action induced by $U_{T}$ on the quotient Banach space $C(X)/\overline{B_{T}(X)}$ and $\underline{U}_{T}$ for the restriction of $U_{T}$ to the closure $\overline{B_{T}(X)}$ of continuous coboundaries. By a standard application of Hahn-Banach theorem (see e.g. Proposition 2.13 in [Kat01]), a function $\psi$ belongs to $\overline{B_{T}(X)}$ if and only if $\int\psi\,d\mu=0$ for any $\mu\in\mathcal{M}(X,T)$ (resp. $\mu\in\mathcal{M}_{e}(X,T)$ ). It is well-known that unique ergodicity is equivalent to the decomposition $C(X)=\mathbb{R}\mathbb{1}\oplus\overline{B_{T}(X)}$ (see e.g. Lemma 1 in [LV97]), where $\mathbb{1}$ denotes the constant function equal to $1$ . In particular in the case of unique ergodicity, we have $C(X)/\overline{B_{T}(X)}\simeq\mathbb{R}\mathbb{1}$ and therefore $\text{\rm Mult}(\tilde{U}_{T})=1$ . It may be generalized as follows.

Lemma 2.6.

Let $(X,T)$ be an invertible topological dynamical system. We have

\text{\rm Mult}(\tilde{U}_{T})=\sharp\mathcal{M}_{e}(X,T).

Proof.

We first show that $\text{\rm Mult}(\tilde{U}_{T})\geq\sharp\mathcal{M}_{e}(X,T)$ . Assume that :

•

$\nu_{1},\cdots,\nu_{p}$ are distinct ergodic measures,
•

$\overline{F}=\{\overline{f_{1}},\cdots,\overline{f_{q}}\}\in C(X)/\overline{B_{T}(X)}$ is a generating family of $\tilde{U}_{T}$ .

For $1\leq l\leq q$ , let $f_{l}\in C(X)$ be a function (a priori not unique) such that $\overline{f_{l}}=f_{l}\mod\overline{B_{T}(X)}$ . If $q<p$ then the $p$ vectors

X_{i}=\left(\int f_{l}\,d\nu_{i}\right)_{l=1,2,\cdots,q},i=1,2,\cdots,p

are linearly dependent in $\mathbb{R}^{q}$ , i.e. there is $(c_{i})_{1\leq i\leq p}\in\mathbb{R}^{p}\setminus(0,0,\dots,0)$ such that

\cdot

\displaystyle\sum_{1\leq i\leq p}c_{i}X_{i}

\displaystyle=0.

Let $\nu$ be the signed measure $\nu=\sum_{1\leq i\leq p}c_{i}\nu_{i}$ . Then Equality (2 $\cdot$ 2) may be rewritten as follows:

\forall 1\leq l\leq q,\ \int f_{l}\,d\nu=0.

The measures $\nu_{i}$ being invariant for ${1\leq i\leq p}$ , so is $\nu$ . Therefore we get

\cdot

\forall 1\leq l\leq q,\forall k\in\mathbb{Z},\ \int f_{l}\circ T^{k}\,d\nu=0.

But $V^{\tilde{U}_{T}}_{\overline{F}}=C(X)/\overline{B_{T}(X)}$ , so that for any $\epsilon>0$ and for any $g\in C(X)$ , we may find $h\in\mathrm{span}(f_{l}\circ T^{k},\ 1\leq l\leq q,k\in\mathbb{Z})$ and $u\in B_{T}(X)$ with $\|g-(h+u)\|_{\infty}<\epsilon$ . By 2 $\cdot$ 3 we have $\int h\,d\nu=0$ . As $u$ is a coboundary, we have also $\int u\,d\nu=0$ . Therefore

\displaystyle\left|\int g\,d\nu\right|

\displaystyle\leq\left|\int(h+u)\,d\nu\right|+\|g-(h+u)\|_{\infty}<\epsilon.

Since $\epsilon>0$ and $g\in C(X)$ are chosen arbitrarily, we obtain $\int g\,d\nu=0$ , for any $g\in C(X)$ , therefore $\nu=0$ . This contradicts the ergodicity of the measures $\nu_{i}$ for $1\leq i\leq p$ . Consequently we have $q\geq p$ and therefore $\text{\rm Mult}(\tilde{U}_{T})\geq\sharp\mathcal{M}_{e}(X,T)$ .

Let us show now the converse inequality. Without loss of generality we may assume that $p=\sharp\mathcal{M}_{e}(X,T)<\text{\rm Mult}(\tilde{U}_{T})=q<\infty$ . We let again:

•

$\mathcal{M}_{e}(X,T)=\{\nu_{1},\cdots,\nu_{p}\}$ ,
•

$\overline{F}={\overline{f_{1}},\cdots,\overline{f_{q}}}\in C(X)/\overline{B_{T}(X)}$ a generating family of $\tilde{U}_{T}$ with minimal cardinality.

Then the $q$ vectors

Y_{l}=\left(\int f_{l}\,d\nu_{i}\right)_{i=1,\cdots,p},\ l=1,\cdots,q,

are linearly dependent in $\mathbb{R}^{p}$ , i.e. there is $(c_{l})_{1\leq l\leq q}\in\mathbb{R}^{q}\setminus(0,0\dots,0)$ such that

\sum_{1\leq l\leq q}c_{l}Y_{l}=0.

Let $g$ be the function $g=\sum_{1\leq l\leq q}c_{l}f_{l}$ . Then we have

\int g\,d\nu_{i}=0,\ \forall 1\leq i\leq p,

A previously mentioned, it implies that $g$ lies in $\overline{B_{T}(X)}$ . This contradicts the minimality of the generating family $\overline{F}$ . ∎

Remark 2.7.

It follows from the proof of Lemma 2.6 that if $\mathcal{M}_{e}(X,T)=\{\nu_{1},\cdots,\nu_{p}\}$ , then ${\overline{f_{1}},\cdots,\overline{f_{p}}}$ is a generating family of $\tilde{U}_{T}$ if and only if the matrix $A=(\int f_{j}\,d\nu_{i})_{1\leq i,j\leq p}\in M_{p}(\mathbb{R})$ is invertible.

By inequality (2 $\cdot$ 1) and Lemma 2.6 we get:

Corollary 2.8.

\sharp\mathcal{M}_{e}(X,T)\leq\text{\rm Mult}(T).

2.4. Relating $\text{\rm Mult}(T)$ and $\text{\rm Mult}(\underline{U}_{T})$

It follows from definition of $\overline{B_{T}(X)}$ that the map

W:C(X)\rightarrow\overline{B_{T}(X)},\ f\mapsto f\circ T-f

has dense image and commutes with $U_{T}$ . By applying Lemma 2.1 with $U_{1}=U_{2}=U$ and $E_{1}=C(X)$ , $E_{2}=\overline{B_{T}(X)}$ , we obtain $\text{\rm Mult}(\underline{U}_{T})\leq\text{\rm Mult}(T)$ .

We show then in this subsection the following inequality.

Proposition 2.9.

\text{\rm Mult}(T)\leq\text{\rm Mult}(\tilde{U_{T}})+\text{\rm Mult}(\underline{U}_{T})-1.

In particular if $(X,T)$ is uniquely ergodic, $\text{\rm Mult}(T)=\text{\rm Mult}(\underline{U}_{T})$ by Lemma 2.6. Let us now prove Proposition 2.9. For a family $F$ of $C(X)$ , we write $\overline{F}$ the subset of $C(X)/\overline{B_{T}(X)}$ consisting of $\overline{f}=f\mod\overline{B_{T}(X)}$ for $f\in F$ . We start with a technical lemma.

Lemma 2.10.

Let $(X,T)$ be an invertible dynamical system with $\sharp\mathcal{M}_{e}(X,T)<\infty$ . If $F$ is a family of $C(X)$ such that $\overline{F}$ is generating for $\tilde{U}_{T}$ , then the constant function $\mathbb{1}$ belongs to $V_{F}$ .

Proof.

Let $\mathcal{M}_{e}(X,T)=\{\nu_{1},\cdots,\nu_{p}\}$ . By Remark 2.7 the matrix $(\int f_{i}\,d\nu_{j})_{1\leq i,j\leq p}$ is invertible. Then by replacing $F=\{f_{1},\cdots,f_{p}\}$ by some invertible linear combinations we can assume

\forall 1\leq i,j\leq p,\ \int f_{i}\,d\nu_{j}=\delta_{i,j},

where $\delta_{i,j}$ is equal to $1$ if $i=j$ and $0$ otherwise. Let $f=\sum_{i=1}^{p}f_{i}$ . We have

	$\displaystyle\forall 1\leq j\leq p,$	$\displaystyle\int f\,d\nu_{j}=\sum_{i}\delta_{i,j}=1,$
(2 $\cdot$ 4)		$\displaystyle\text{therefore, }\forall\nu\in\mathcal{M}(X,T),$	$\displaystyle\int f\,d\nu=1.$

We claim that $\frac{1}{N}\sum_{n=0}^{N-1}f\circ T^{n}$ is converging uniformly to $\mathbb{1}$ as $N$ goes to infinity. If not, there would exist a positive number $\epsilon$ , a sequence $(x_{k})_{k\geq 1}$ and an increasing sequence $(N_{k})_{k\geq 1}$ of positive integers such that

\cdot

\left|\frac{1}{N_{k}}\sum_{n=0}^{N_{k}-1}f(T^{n}(x_{k}))-1\right|>\epsilon,\ \forall k\geq 1.

After passing to a subsequence of $(N_{k})_{k\geq 1}$ , we might assume that $\frac{1}{N_{k}}\sum_{n=0}^{N_{k}-1}\delta_{T^{n}(x_{k})}$ is converging to a $T$ -invariant measure $\mu$ in the weak- $*$ topology. It follows from (2 $\cdot$ 5) that

\left|\int fd\mu-1\right|>\epsilon>0.

It is a contradiction to (2.4). Therefore, $\frac{1}{N}\sum_{n=0}^{N-1}f\circ T^{n}$ is converging uniformly to $\mathbb{1}$ as $N$ goes to infinity, in particular $\mathbb{1}\in V_{F}$ . ∎

Proof of Proposition 2.9.

Let $\overline{\mathcal{F}}=\{\overline{f_{1}},\cdots,\overline{f_{p}}\}$ and $\mathcal{G}=\{g_{1},\cdots,g_{q}\}$ be generating families of $\tilde{U}_{T}$ and $\underline{U}_{T}$ with $p=\text{\rm Mult}(\tilde{U}_{T})$ and $q=\text{\rm Mult}(\underline{U}_{T})$ . For $1\leq l\leq p$ , take $f_{l}\in C(X)$ be a function such that $\overline{f_{l}}=f_{l}\mod\overline{B_{T}(X)}$ , then let $\mathcal{F}=\{f_{1},\cdots,f_{l}\}$ . One easily checks that $\mathcal{F}\cup\mathcal{G}$ is a generating family of $U_{T}$ . By Lemma 2.6 we may write $\mathcal{M}_{e}(X,T)=\{\nu_{1},\cdots,\nu_{p}\}$ . As in the proof of Lemma 2.10 we may assume without loss of generality $\int f_{i}\,d\nu_{j}=\delta_{i,j}$ for any $1\leq i,j\leq p$ . Let $g^{\prime}_{1}=g_{1}+\mathbb{1}$ , hence

\left(\int g^{\prime}_{1}\,d\nu_{i}\right)_{1\leq i\leq p}=(1,\cdots,1).

By Remark 2.7 the family $\{\overline{g^{\prime}_{1}},\overline{f_{j}}:\ 1<j\leq p\}$ is generating for $\tilde{U}_{T}$ . By Lemma 2.10 the constant functions, therefore also $g_{1}$ , belongs to $V_{\{g^{\prime}_{1},f_{j}:\ 1<j\leq p\}}$ . Then $V_{\{g^{\prime}_{1},f_{j},g_{i}\ :\ 1<j\leq p,\,1<i\leq q\}}=V_{\{g^{\prime}_{1},f_{j},g_{i}\ :\ 1<j\leq p,\,1\leq i\leq q\}}$ and $f_{1}\in V_{\{g^{\prime}_{1},f_{j},g_{i}\ :\ 1<j\leq p,\,1\leq i\leq q\}}$ . Consequently we get

\displaystyle V_{\{g^{\prime}_{1},f_{j},g_{i}\ :\ 1<j\leq p,\,1<i\leq q\}}

\displaystyle\supset V_{\mathcal{F}\cup\mathcal{G}}=C(X).

We conclude that $\text{\rm Mult}(T)\leq p+q-1=\text{\rm Mult}(\tilde{U_{T}})+\text{\rm Mult}(\underline{U}_{T})-1.$ ∎

3. Affine embedding of $(\mathcal{M}(X),T_{*})$ in cubical shifts

For a topological system $(X,T)$ we denote by $T_{*}$ the action induced by $T$ on the compact set $\mathcal{M}(X)$ , i.e. $T_{*}\mu(\cdot)=\mu(T^{-1}\cdot)$ for all $\mu\in\mathcal{M}(X)$ . Then $(\mathcal{M}(X),T_{*})$ is also a topological system, which is called the induced system of $(X,T)$ .

For $d\in\mathbb{N}$ we let $\sigma_{d}$ be the shift on the simplex $([0,1]^{d})^{\mathbb{Z}}$ . An embedding of $(X,T)$ in $([0,1]^{d})^{\mathbb{Z}}$ is a continuous injective map $\phi:X\rightarrow([0,1]^{d})^{\mathbb{Z}}$ satisfying $\phi\circ T=\sigma_{d}\circ\phi$ . Existence of such embedding is related to the mean dimension theory (we refer to [Coo15] for an introduction). Such an embedding implies that the mean dimension of $(X,T)$ is less than or equal to $d$ . Moreover the topological dimension (i.e. Lebesgue covering dimension) $d_{n}^{T}$ of the set of $n$ -periodic points then also satisfy $\frac{d_{n}^{T}}{n}\leq d$ . Conversely it has been shown that minimal systems with mean dimension less than $d/2$ can be embedded in the cubical shift $\sigma_{d}$ [Lin99, GT20].

In this section we consider affine embedding of the induced system $(\mathcal{M}(X),T_{*})$ in cubical shift $\sigma_{d}$ , i.e. the embedding $\phi:\mathcal{M}(X)\rightarrow([0,1]^{d})^{\mathbb{Z}}$ is affine. In particular we will relate the embedding dimension $d$ with the multiplicity of $(X,T)$ .

3.1. Case of finite sets

We first deal with the case of a finite set $X$ . Then $T$ is just a permutation of $X$ and $\mathcal{M}(X)$ is a finite dimensional simplex. We classify the possible affine embedding of $(\mathcal{M}(X),T_{*})$ in the following proposition.

Proposition 3.1.

Suppose $X$ is a finite set and $T$ is a a permutation of $X$ . Let $\tau_{1}\cdots\tau_{k}$ be the decomposition of $T$ into disjoint cycles $\tau_{i}$ of length $r_{i}$ for $1\leq i\leq k$ .

(1)

If there is a nontrivial common factor of $r_{i}$ for $1\leq i\leq k$ , then there is an affine embedding of $(\mathcal{M}(X),T_{*})$ in $\left(([0,1]^{k})^{\mathbb{Z}},\sigma_{k}\right)$ . Such $k$ is sharp.
(2)

If there is no nontrivial common factor of $r_{i}$ for $1\leq i\leq k$ , then there is an affine embedding of $(\mathcal{M}(X),T_{*})$ in $\left(([0,1]^{k-1})^{\mathbb{Z}},\sigma_{k-1}\right)$ . Such $k-1$ is sharp.

Proof.

For each $1\leq i\leq k$ we fix a point $e_{i}\in X$ in each cycle $\tau_{i}$ , i.e. $\{T^{j}e_{i}:0\leq j\leq r_{i}\}=X$ . Notice that there are continuous maps $a_{e}:\mathcal{M}(X)\rightarrow[0,1]$ , $e\in X$ , with $\sum_{e\in X}a_{e}=\mathbb{1}$ satisfying $\mu=\sum_{e\in X}a_{e}(\mu)\delta_{e}$ for all $\mu\in\mathcal{M}(X)$ .
(1) Assume there is a nontrivial common factor $p$ of $r_{i}$ for $1\leq i\leq k$ . Then

\mathrm{dim}(\mathrm{Fix}(T_{*}^{p}))=kp-1,

where $\mathrm{Fix}(T_{*}^{p})=\{\mu\in\mathcal{M}(X):\ T_{*}^{p}\mu=\mu\}$ . Since $p>1$ and $\mathrm{dim}(\mathrm{Fix}(\sigma_{k-1}^{p}))=kp-p$ , the dynamical system $(\mathcal{M}(X),T_{*})$ can not embed in $\left(([0,1]^{k-1})^{\mathbb{Z}},\sigma_{k-1}\right)$ .

Now we construct the embedding of $(\mathcal{M}(X),T_{*})$ in $\left(([0,1]^{k})^{\mathbb{Z}},\sigma_{k}\right)$ . We define firstly a dynamical embedding $\Psi$ of the set of extreme points in $\mathcal{M}(X)$ , which is identified with $X$ through the map $x\mapsto\delta_{x}$ , into $([0,1]^{k})^{\mathbb{Z}}$ by letting

\forall i=1,\cdots,k,\ \forall l\in\mathbb{Z},\ \ (\Psi(T^{l}e_{i}))_{i}=\sigma^{l}\left((10^{r_{i}-1})^{\infty}\right);

the other components $(\Psi(T^{l}e_{i}))_{j}$ , $j\neq i$ , being chosen to be equal to the $0^{\infty}$ sequence. Then we may extend $\Psi$ affinely from the set of extreme points on $\mathcal{M}(X)$ by letting

\Psi(\mu)=\sum_{e\in X}a_{e}(\mu)\Psi(\delta_{e}).

It is easy to check that $\Psi$ is injective which deduces a dynamical embedding $(\mathcal{M}(X),T_{*})$ in $\left(([0,1]^{k})^{\mathbb{Z}},\sigma_{k}\right)$ .

(2) Assume there is no nontrivial common factor of $r_{i}$ for $1\leq i\leq k$ . We have that $k-1$ numbers $q_{i}:=(r_{k},r_{i}),$ $1\leq i\leq k-1$ are co-prime where $(a,b)$ are the highest common factor of $a$ and $b$ . We define firstly a continuous map $\Psi$ of the set of extreme points in $\mathcal{M}(X)$ into $([0,1]^{k-1})^{\mathbb{Z}}$ by letting

\ \forall l\in\mathbb{Z},\ \ (\Psi(T^{l}e_{k}))_{j}=\sigma^{l}\left((10^{r_{i}-1})^{\infty}\right),\forall 1\leq j\leq k-1,

and

\forall 1\leq i\leq k-1,\ \forall l\in\mathbb{Z},\ \ (\Psi(T^{l}e_{i}))_{i}=\sigma^{l}\left((10^{r_{i}-1})^{\infty}\right);\ \forall j\not=i,\forall l\in\mathbb{Z},(\Psi(T^{l}e_{i}))_{j}=0^{\infty}.

Then we may extend $\Psi$ affinely from the set of extreme points on $\mathcal{M}(X)$ by letting

\Psi(\mu)=\sum_{e\in X}a_{e}(\mu)\Psi(\delta_{e}).

It remains to show that $\Psi$ is injective. Let $\mu=\sum_{e\in X}b_{e}\delta_{e}$ and $\mu^{\prime}=\sum_{e\in X}b_{e}^{\prime}\delta_{e}$ . Suppose

\Psi(\mu)=\Psi(\mu^{\prime})=\sum_{e\in X}a_{e}\Psi(\delta_{e})=(c_{i,j})_{1\leq i\leq k-1,j\in\mathbb{Z}}.

Since $q_{i}:=(r_{k},r_{i})$ then there are integers $s_{i}$ and $t_{i}$ such that $s_{i}r_{i}-t_{i}r_{k}=q_{i}.$ Let

u_{i,l}=s_{i}r_{i}+l=t_{i}r_{k}+q_{i}+l.

It implies that

b_{T^{l}e_{k}}^{\prime}-b_{T^{l+q_{i}}e_{k}}^{\prime}=b_{T^{l}e_{k}}-b_{T^{l+q_{i}}e_{k}}=c_{i,l}-c_{i,u_{i,l}}.

Since $q_{i},1\leq i\leq k-1$ are co-prime, there are integers $w_{i},1\leq i\leq k-1$ such that $\sum_{i=1}^{k-1}w_{i}q_{i}=1$ . Since

	$\displaystyle\left(b_{e_{k}}-b_{T^{w_{1}q_{1}}e_{k}}\right)+$	$\displaystyle\left(b_{T^{w_{1}q_{1}}e_{k}}-b_{T^{w_{1}q_{1}+w_{2}q_{2}}e_{k}}\right)$
		$\displaystyle+\dots+\left(b_{T^{w_{1}q_{1}+w_{2}q_{2}+\dots+w_{k-2}q_{k-2}}e_{k}}-b_{T^{w_{1}q_{1}+w_{2}q_{2}+\dots+w_{k-1}q_{k-1}}e_{k}}\right)=b_{e_{k}}-b_{Te_{k}},$

we have

b_{e_{k}}-b_{T^{l}e_{k}}=b_{e_{k}}^{\prime}-b_{T^{l}e_{k}}^{\prime},\ \forall l\in\mathbb{Z}.

Since $\sum_{e\in X}b_{e}=\sum_{e\in X}b_{e}^{\prime}=1$ , we conclude that $b_{e_{k}}=b_{e_{k}^{\prime}}$ and consequently $b_{e_{i}}=b_{e_{i}^{\prime}}$ by $b_{e_{k}}+b_{e_{i}}=b_{e_{k}}^{\prime}+b_{e_{i}}^{\prime}=c_{i,0}$ for $1\leq i\leq k-1$ . It means that $\mu=\mu^{\prime}$ and $\Psi$ is injective.

∎

Remark 3.2.

For such a permutation $T$ , we have $\text{\rm Mult}(T)=\sharp\mathcal{M}_{e}(X,T)=k$ , with $k$ being the number of cycles in the decomposition of $T$ .

3.1.1. General case

We consider now a general topological system and relates the dimension of the cubical shift in an affine embedding with the multiplicity of $(X,T)$ .

Theorem 3.3.

Let $(X,T)$ be a topological system. If $\text{\rm Mult}(T)$ is equal to $d$ , then there is an affine embedding of $(\mathcal{M}(X),T_{*})$ in $(([0,1]^{d})^{\mathbb{Z}},\sigma_{d})$ . Conversely if $(\mathcal{M}(X),T_{*})$ embeds into $(([0,1]^{d})^{\mathbb{Z}},\sigma_{d})$ then

•

either $\sharp\mathcal{M}_{e}(X,T)\leq d$ and $\text{\rm Mult}(T)\leq d$ ,
•

or $\sharp\mathcal{M}_{e}(X,T)=d+1$ and $\text{\rm Mult}(T)=d+1$ .

Proof.

Firstly, notice that any affine equivariant map $\Psi:(\mathcal{M}(X),T_{*})\rightarrow(([0,1]^{d})^{\mathbb{Z}},\sigma_{d})$ is of the form

\Psi_{f}:\mu\mapsto\left(\int f\circ T^{k}\,d\mu\right)_{k\in\mathbb{Z}},

for some continuous function $f=(f_{1},\cdots,f_{d}):X\rightarrow[0,1]^{d}$ .

Assume the topological multiplicity $\text{\rm Mult}(X,T)$ is equal to $d$ , i.e. there is a family $F=\{f_{1},\cdots,f_{d}\}$ of continuous functions such that $V_{F}=C(X)$ . Let us show the associated map $\Psi_{f}$ is injective. Let $\mu_{1},\mu_{2}\in\mathcal{M}(X)$ with $\Psi_{f}(\mu_{1})=\Psi_{f}(\mu_{1})$ i.e. $\int f_{i}\circ T^{k}\,d\mu_{1}=\int f_{i}\circ T^{k}\,d\mu_{1}$ for any $i=1,\cdots,d$ and any $k\in\mathbb{Z}$ . Then by density of $\mathrm{span}(f_{i}\circ T^{k},\ i,\ k)$ in $C(X)$ we have

\int g\,d\mu_{1}=\int g\,d\mu_{1}\text{ for all }g\in C(X),

which implies $\mu_{1}=\mu_{2}$ . Therefore we get the injectivity of $\Psi_{f}$ .

Conversely, assume $\Psi_{f}$ is injective for $f=(f_{1},\cdots,f_{d}):X\rightarrow[0,1]^{d}$ . Let $F=\{\mathbb{1},f_{1},\cdots,f_{d}\}$ . We claim that $V_{F}=C(X)$ . Then if $\sharp\mathcal{M}_{e}(X,T)\leq d$ , we get by injectivity of $\Psi_{f}$ that there exists $A\subset\{1,2,\dots,d\}$ with $\sharp A=\sharp\mathcal{M}_{e}(X,T)$ such that the matrix $\left(\int f_{i}d\nu\right)_{i\in A,\nu\in\mathcal{M}_{e}(X,T)}$ is invertible. Then by Remark 2.7 the family $\overline{F\setminus\{\mathbb{1}\}}$ is generating for $\tilde{U}_{T}$ and consequently $V_{F}=V_{F\setminus\{\mathbb{1}\}}$ by Lemma 2.10. If $\sharp\mathcal{M}_{e}(X,T)=d+1$ then we only get $\text{\rm Mult}(T)=d+1$ .

It remains to show our claim. Assume to the contrary that $V_{F}\not=C(X)$ . Then by Riesz Theorem there is a signed finite measure $\mu$ vanishing on each function in $F$ . Let $\mu=\mu^{+}-\mu^{-}$ be the Jordan decomposition of $\mu$ (i.e. the measures $\mu^{+}$ and $\mu^{-}$ are two finite positive measures which are mutually singular). Evaluating on the constant function $\mathbb{1}$ , we get $\mu^{+}(X)=\mu^{-}(X)$ . Then by rescaling, we may assume both $\mu^{-}$ and $\mu^{+}$ belong to $\mathcal{M}(X)$ . Finally we get $\Psi_{f}(\mu^{+})=\Psi_{f}(\mu^{-})$ , and therefore $\mu^{+}=\mu^{-}$ by injectivity of $\Psi_{f}$ contradicting therefore the mutual singularity of $\mu^{+}$ and $\mu^{-}$ .

∎

3.2. Affine embeddings and Lindenstrauss-Tsukamoto conjecture

Lindenstrauss and Tsukamoto [LT14] have conjectured that any topological system with mean dimension $\text{\rm mdim}(X,T)$ less than $d/2$ and such that the dimension $d_{n}^{T}$ of the set of $n$ -periodic points satisfies $\frac{d_{n}^{T}}{n}<d/2$ for any $n\in\mathbb{N}$ may be embedded in the shift over $([0,1]^{d})^{\mathbb{Z}}$ . As mentioned above it is known for minimal systems. We consider here affine systems, i.e. affine maps of a simplex. Such maps are never minimal, as they always admits at least one fixed point.

The example below shows that Lindenstrauss-Tsukamoto conjecture does not hold true in the affine category. Recall that an ergodic system $(X,f,\mathcal{B},\mu)$ has a countable Lebesgue spectrum, when there is a countable family $(\psi_{n})_{n\in\mathbb{N}}$ in $L^{2}_{0}(\mu)$ such that $\psi_{n}\circ f^{k}$ , $k\in\mathbb{Z}$ , $n\in\mathbb{N}$ form a Hilbert basis of $L^{2}_{0}(\mu)$ .

Corollary 3.4.

There is an affine system with a unique periodic (fixed) point and zero-topological entropy (in particular $mdim(T)=0$ and $d_{n}^{T}=0$ for all $n\geq 1$ ) which does not embed affinely in $(([0,1]^{k})^{\mathbb{Z}},\sigma)$ for any $k\geq 1$ .

Proof.

There exists an ergodic measure preserving system $(Y,\mathcal{A},f,\mu)$ with zero entropy and countable Lebesgue spectrum [NP66, Par53] (in particular totally ergodic, i.e. $f^{n}$ is ergodic for any $n\in\mathbb{Z}$ ). Then by Jewett-Krieger theorem there is a uniquely ergodic topological system $(X,T)$ with measure $\nu$ realizing such a measure preserving system. All powers of $T$ are uniquely ergodic as $\mu$ was chosen totally ergodic. Moreover the topological entropy of $T$ , thus that of $T_{*}$ is zero by Glasner-Weiss [GW95]. As the unique invariant measure $\nu$ has countable Lebesgue spectrum, the topological multiplicity of $(X,T)$ is infinite by Lemma 2.4. We conclude with Theorem 3.3. ∎

3.3. Application: zero topological entropy

A classical result in ergodic theory states that any ergodic system $(X,f,\mathcal{B},\mu)$ with positive entropy has a countable Lebesgue spectrum. In particular $[h(\mu)>0]\Rightarrow[\text{\rm Mult}(\mu)=\infty]$ . Then it follows from the variational principle for the topological entropy :

Proposition 3.5.

Any topological system $(X,T)$ with $\text{\rm Mult}(T)<\infty$ has zero topological entropy.

We may also give a purely topological proof of Proposition 3.5 based on mean dimension theory. More precisely we use the main result of [BS22], which states as follows:

Theorem 3.6.

[BS22] For any topological system $(X,T)$ with positive topological entropy, the induced system $(\mathcal{M}(X),T_{*})$ has infinite topological mean dimension. Therefore,

h_{top}(T)>0\Leftrightarrow\text{\rm mdim}(T_{*})>0\Leftrightarrow\text{\rm mdim}(T_{*})=\infty.

Topological proof of Proposition 3.5.

Assume $\text{\rm Mult}(T)=d$ is finite. Then by Theorem 3.3 the induced system $(\mathcal{M}(X),T_{*})$ embeds in the cubical shift $\left(([0,1]^{d})^{\mathbb{Z}},\sigma\right)$ . In particular the mean dimension of $T_{*}$ is less than or equal to the mean dimension of the shift $\left(([0,1]^{d})^{\mathbb{Z}},\sigma\right)$ , which is equal to $d$ . By Theorem 3.6, it implies that $T$ has zero topological entropy. ∎

4. Baxter’s Lemma in Banach spaces

In [Bax71], Baxter gave a useful criterion to show simple spectrum of ergodic transformations. It may be extended more generally to bound the multiplicity of the spectrum (e.g. see Proposition 2.12 in [Que10]). We generalize this criterion for operators defined on a Banach space. It will be used in the next section to estimate the topological multiplicity in some examples.

Lemma 4.1.

Let $B$ be a separable Banach space and $\mathcal{L}(B)$ be the set of bounded linear operator on $B$ . We consider an invertible isometry $U\in\mathcal{L}(B)$ . If $(\mathcal{F}_{n})_{n}$ is a sequence of finite subsets in $H$ satisfying for all $f\in B$

\cdot

\displaystyle\inf_{F_{n}\in V_{\mathcal{F}_{n}}}\|F_{n}-f\|\xrightarrow{n\rightarrow\infty}0

then there exists a family $\mathcal{F}\subset B$ with $\sharp\mathcal{F}\leq\sup_{n}\sharp\mathcal{F}_{n}$ and $B=V_{\mathcal{F}}.$

Classical proofs of Baxter’s lemma strongly used the Hilbert structure. Here we use a Baire argument as in Lemma 5.2.10 [Fog02]. Note also that we do not require the sequence of vectors spaces $(V_{\mathcal{F}_{n}})_{n}$ to be nondecreasing. Observe finally that it is enough to assume (4 $\cdot$ 1) for $f$ in $S$ , where $S$ spans a dense subset of $B$ .

Proof.

Let $m=\sup_{n}\sharp\mathcal{F}_{n}$ . If $m=\infty$ , it is trivial. Assume $m<\infty$ . By passing to a subsequence, we assume $\sharp\mathcal{F}_{n}=m$ for all $n$ . Let $B^{(m)}$ be the space of finite subsets of $H$ whose cardinality is smaller than or equal to $m$ . When endowed with the Hausdorff distance $d_{Hau}$ , the space $B^{(m)}$ is a metric space, which is complete and separable. We assume the following claim, which we prove later on.

Claim 4.2.

For any $\epsilon>0$ and $\mathcal{F}\in B^{(m)}$ , the set

O(\mathcal{F},\epsilon)=\left\{\mathcal{G}\in B^{(m)},\ \forall f\in\mathcal{F}\ \inf_{G\in V_{\mathcal{G}}}\|G-f\|<\epsilon\right\}

is open and dense.

Let $\left(g_{q}\right)_{q\in\mathbb{N}}$ be a countable dense family in $B$ . Let $\mathcal{G}_{q}$ be the finite family $\{g_{1},\cdots,g_{q}\}$ . For any $p\in\mathbb{N}^{*}$ and any $q\in\mathbb{N}$ we consider the open and dense set

O_{p,q}:=O(\mathcal{G}_{q},1/p)=\left\{\mathcal{G}\in B^{(m)},\ \forall g\in\mathcal{G}_{q}\ \inf_{G\in V_{\mathcal{G}}}\|G-g\|<1/p\right\}.

According to Baire’s theorem, the intersection $\bigcap_{p,q}O_{p,q}$ is not empty. Clearly any family $\mathcal{F}$ in the intersection satisfies $V_{\mathcal{F}}=B$ . It remains to prove Claim 4.2.

Proof of Claim 4.2.

The set $O(\mathcal{F},\epsilon)$ is open. We focus on the denseness property. Pick arbitrary $\delta>0$ and $\mathcal{H}\in B^{(m)}$ . We will show that there is $\mathcal{H}^{\prime}\in O(\mathcal{F},\epsilon)$ with $d_{Hau}(\mathcal{H},\mathcal{H}^{\prime})<\delta$ . As the elements of $B^{(m)}$ with cardinality $m$ are dense in $B^{(m)}$ we can assume without loss of generality that $\sharp\mathcal{H}=m$ . By assumptions on the sequence $\mathcal{F}_{n}$ , there exists $n$ such that

\cdot

\displaystyle\forall f\in\mathcal{F}\ \inf_{F_{n}\in V_{\mathcal{F}_{n}}}\|F_{n}-f\|<\epsilon,

\cdot

\displaystyle\forall h\in\mathcal{H}\ \inf_{F_{n}\in V_{\mathcal{F}_{n}}}\|F_{n}-h\|<\delta.

We write $\mathcal{H}=\{h_{1},h_{2},\dots,h_{m}\}$ and $\mathcal{F}_{n}=\{f_{1},f_{2},\dots,f_{m}\}$ . By (4 $\cdot$ 3) there are polynomials $(P_{i,j})_{1\leq i,j\leq m}$ in $\mathbb{R}[X]$ and a nonnegative integer $p$ such that

\cdot

\displaystyle\forall i=1,\cdots,m,\ \left\|h_{i}-\sum_{j=1}^{m}U^{-p}P_{i,j}(U)f_{j}\right\|<\delta

Let $Q\in\mathbb{R}[X]$ be the polynomial given by the determinant of the matrix $M=(P_{i,j})_{1\leq i,j\leq m}\in M_{m}(\mathbb{R}[X])$ . The spectrum $\mathrm{Sp}(U)$ of $U$ is contained in the unit circle. In particular for arbitrarily small $\lambda\in\mathbb{R}$ , the polynomial $Q(\cdot+\lambda)$ does not vanish on $\mathrm{Sp}(U)$ . Hence by replacing $P_{i,j}$ by $P_{i,j}(\cdot+\lambda)$ we may assume that $Q$ does not vanish on the spectrum of $U$ . Then $Q(U)=\prod_{\lambda,\,Q(\lambda)=0}(U-\lambda\mathrm{Id})$ is invertible and its inverse may be approximated by polynomials in $U$ and $U^{-1}$ , because for $\lambda$ with $Q(\lambda)=0$ we have $(U-\lambda\mathrm{Id})^{-1}=-\sum_{k\in\mathbb{N}}\frac{U^{k}}{\lambda^{k+1}}$ for $|\lambda|>1$ and $(U-\lambda\mathrm{Id})^{-1}=-\sum_{k\in\mathbb{N}}\frac{U^{-(k+1)}}{\lambda^{k}}$ for $|\lambda|<1$ (these sequences are normally convergent in $\mathcal{L}(B)$ as we assume $\|U\|=\|U^{-1}\|=1$ ). Let $\mathcal{H}^{\prime}=\{h^{\prime}_{1},h^{\prime}_{2},\dots,h^{\prime}_{m}\}$ with $h^{\prime}_{i}=\sum_{j=1}^{m}U^{-p}P_{i,j}(U)f_{j}$ . We have

	$\displaystyle\mathcal{F}_{n}$	$\displaystyle=M(U)^{-1}U^{p}\mathcal{H}^{\prime},$
		$\displaystyle=\mathrm{{}^{t}com}\,M(U)U^{p}Q(U)^{-1}\mathcal{H}^{\prime}.$

Then from the above observations we get

\mathcal{F}_{n}\subset V_{\mathcal{H}^{\prime}}.

In particular $V_{\mathcal{F}_{n}}\subset V_{\mathcal{H}^{\prime}}$ and it follows finally from (4 $\cdot$ 2) that $\mathcal{H}^{\prime}\in O(\mathcal{F},\epsilon)$ , i.e.

\displaystyle\forall f\in\mathcal{F}\ \inf_{H^{\prime}\in V_{\mathcal{H}^{\prime}}}\|H^{\prime}-f\|<\epsilon.

This completes the proof as we have $d_{Hau}(\mathcal{H},\mathcal{H}^{\prime})<\delta$ by (4 $\cdot$ 4), where $\mathcal{H}$ and $\delta$ have been chosen arbitrarily. ∎

∎

5. Cantor systems with finite topological rank

Roughly speaking an ergodic measure preserving system is of finite rank $r$ , when it may be obtained by cutting and staking with $r$ Kakutani-Rohlin towers. For ergodic systems, Baxter’s lemma implies that the ergodic multiplicity is less than or equal to the rank. Topological rank as been defined and studied for minimal Cantor systems (see e.g. [DDMP21] and the references therein). For such systems we show now with Lemma 4.1 that the same inequality holds for the topological quantities : the topological multiplicity is less than or equal to the topological rank.

Firstly we recall the definition of topological rank. Let $(X,T)$ be a minimal Cantor system. A Kakutani-Rohlin partition of $X$ is given by

\mathcal{T}=\{T^{-j}B(k):1\leq k\leq d,0\leq j<h(k)\},

where $d,h(k),1\leq k\leq d$ are positive integers and $B(k),1\leq k\leq d$ are clopen subsets of $X$ such that

\cup_{k=1}^{d}T^{-h(k)}B(k)=\cup_{k=1}^{d}B(k).

The base of $\mathcal{T}$ is the set $B(\mathcal{T})=\cup_{k=1}^{d}B(k)$ . A sequence of Kakutani-Rohlin partitions

\mathcal{T}_{n}=\{T^{-j}B_{n}(k):1\leq k\leq d_{n},0\leq j<h_{n}(k)\},n\geq 1,

is nested if

(1)

$\mathcal{T}_{0}$ is the trivial partition, i.e. $d_{0}=1,h_{0}=1$ and $B_{0}(1)=X$ .
(2)

$B(\mathcal{T}_{n+1})\subset B(\mathcal{T}_{n})$ .
(3)

$\mathcal{T}_{n+1}\succ\mathcal{T}_{n}$ .
(4)

$\sharp(\cap_{n\geq 0}B(\mathcal{T}_{n}))=1$ .
(5)

$\cup_{n\geq 1}\mathcal{T}_{n}$ spans the topology of $X$ .

Moreover, it is primitive if for all $n\geq 1$ there exists $N>n$ such that for all $1\leq k\leq d_{N}$ and for each $x\in T^{-(hN(k)-1)}B_{N}(k)$ ,

\{T^{i}(x):0\leq i\leq h_{N}(k)-1\}\cap B_{n}(j)\not=\emptyset,\forall 1\leq j\leq d_{n}.

Following [DDMP21], a minimal Cantor system is of topological rank $d$ if it admits a primitive sequence of nested Kakutani-Rohlin partitions with $d_{n}\leq d$ for all $n\in\mathbb{N}$ .

Theorem 5.1.

Let $(X,T)$ be a minimal Cantor system with topological rank $d$ . Then $\text{\rm Mult}(X,T)\leq d$ .

Proof.

Let $(\mathcal{T}_{n})_{n\in\mathbb{N}}$ be the primitive sequence of nested Kakutani-Rohlin partitions with $d_{n}\leq d$ for all $n\in\mathbb{N}$ . Let

\mathcal{F}_{n}=\{\chi_{B_{n}(k)}:1\leq k\leq d_{n}\}.

Since $\cup_{n\geq 1}\mathcal{T}_{n}$ spans the topology of $X$ , we have

\forall f\in C(X),\ \inf_{F_{n}\in V_{\mathcal{F}_{n}}}\|F_{n}-f\|\xrightarrow{n\rightarrow\infty}0.

It follows from Lemma 4.1 and $d_{n}\leq d$ for all $n\in\mathbb{N}$ that $\text{\rm Mult}(X,T)\leq d$ . ∎

Remark 5.2.

It was shown in [DDMP21] that the Thue-Morse subshift has topological rank $3$ . By Theorem 5.1, Thue-Morse subshift has therefore topological multiplicity at most $3$ . We give another direct proof of this fact in Subsection 6.4.

Examples of ergodic systems with rank $r$ and multiplicity $m$ have been built for any $1\leq m\leq r$ in [KL97]. We then propose the following question.

Question 5.3.

Can one build for any $1\leq m\leq r$ a minimal Cantor system with topological multiplicity $m$ and topological rank $r$ ?

6. Examples of finite topological multiplicity

An invertible dynamical system is called topological simple or have simple topological spectrum if $\text{\rm Mult}(T)=1$ .

6.1. Minimal rotation on compact groups

Let $G$ be a compact abelian group. Denote by $\hat{G}$ the dual group of $G$ and by $\lambda$ the Haar measure on $G$ . For $f\in C(G)$ , we write $\hat{f}$ the Fourier transformation of $f$ .

Proposition 6.1.

Any minimal translation $\tau$ on a compact abelian group $G$ is topologically simple.

Proof.

We claim that any $f\in C(G)$ with $\hat{f}(\chi)\neq 0$ for all $\chi\in\hat{G}$ is cyclic, i.e. the vector space spanned by $f\circ\tau^{k}$ , $k\in\mathbb{N}$ is dense in $C(X)$ . As characters of a compact abelian group separates points it is enough to show by Stone-Weierstrass theorem that any character belongs to the complex vector space spanned by $f\circ\tau^{k}$ , $k\in\mathbb{N}$ . But for all $\chi\in\hat{G}$ we have

\hat{f}(\chi)\chi=\chi*f=\int f(\cdot-y)\chi(y)\,d\lambda(y).

Then the function $f$ being uniformly continuous, there are functions of the form $\sum_{k}f(\cdot-y_{k})\chi(y_{k})$ arbitrarily close to $\hat{f}(\chi)\chi$ for the supremum norm. By minimality of $\tau$ , there are integers $l_{k}\in\mathbb{N}$ such that $f\circ\tau^{l_{k}}$ and $f(\cdot-y_{k})$ are arbitrarily closed. It concludes the proof. ∎

6.2. Sturmian subshift

A word $u\in\{0,1\}^{\mathbb{Z}}$ is called Sturmian if it is recurrent under the shift $\sigma$ , and the number of $n$ -words in $u$ equals $n+1$ for each $n\geq 1$ . Take the shift-orbit closure $X_{u}=\overline{O_{\sigma}(u)}$ . The corresponding subshift $(X_{u},\sigma)$ is called a Sturmian subshift. Sturmian sequences are symbolic representation of circle irrational rotations.

We first recall some standard notations in symbolic dynamics. For a subset $Y$ of $\mathcal{A}^{\mathbb{Z}}$ with $\mathcal{A}$ being a finite alphabet we let $\mathcal{L}_{n}(Y)$ be the number of $n$ -words appearing in the sequences of $Y$ . Then for $w\in\mathcal{L}_{n}(Y)$ we let $[w]$ be the associated cylinder defined as $[w]:=\{(x_{n})_{n\in\mathbb{Z}}\in Y:\ x_{0}\cdots x_{n-1}=w\}.$ The indicator function of a subset $E$ of $X$ will be denoted by $\chi_{E}$ .

Proposition 6.2.

Any Sturmian subshift has simple topological spectrum.

Proof.

Let $u$ be a Sturmian sequence. Let

F_{n}=\text{span}\{\chi_{[w]}:w\in\mathcal{L}_{n}(u)\}.

It follows that

C(X_{u})=V_{\cup_{n}F_{n}}.

Notice $\dim(F_{n})=\sharp\mathcal{L}_{n}(u)=n+1$ . We let $f:X_{u}\rightarrow\mathbb{R}$ be the continuous function defined as $f:x=(x_{n})_{n}\mapsto(-1)^{x_{0}}$ . Let

G_{n}=\text{span}\{\mathbb{1},f\circ\sigma^{k}:0\leq k\leq n-1\}.

Clearly, $G_{n}\subset F_{n}$ . To prove that $(X_{u},\sigma)$ has simple topological spectrum, it is sufficient to show $\dim(G_{n})=n+1$ . Thus it is enough to show the functions $\{\mathbb{R}\mathbb{1},f\circ\sigma^{k}:0\leq k\leq n-1\}$ are linearly independent. If not, for some $n$ there exists a nonzero vector $(a_{0},a_{1},\dots,a_{n})$ such that

a_{0}(-1)^{x_{0}}+a_{1}(-1)^{x_{1}}+\dots+a_{n-1}(-1)^{x_{n-1}}+a_{n}=0,\forall x\in X_{u}.

Since $\sharp\mathcal{L}_{n-1}(u)>\sharp\mathcal{L}_{n-2}(u)$ , we can find distinct $x,x^{\prime}\in X_{u}$ such that $x|_{0}^{n-2}=x^{\prime}|_{0}^{n-2}$ but $x_{n-1}\not=x^{\prime}_{n-1}$ . It follows that $a_{n-1}=0$ . Since for each $0\leq k\leq n-2$ we can always find $y,y^{\prime}$ such that $y|_{0}^{k-1}=y^{\prime}|_{0}^{k-1}$ but $y_{k}\not=y^{\prime}_{k}$ , we obtain that $a_{n-2}=a_{n-3}=\dots=a_{0}=0$ . Finally, we get $a_{n}=0$ . This is a contradiction. Therefore we conclude that $\dim(G_{n})=n+1$ , then $F_{n}=G_{n}$ . Then by Lemma 4.1 and Lemma 2.10, $(X_{u},\sigma)$ has simple topological spectrum. ∎

6.3. Homeomorphism of the interval

Estimating the multiplicity of non-zero dimensional systems is difficult in general. Below we focus on homeomorphisms of the interval (see [Jav19] for related results on the circle). We use the following result due to Atzmon and Olevskii [AO96]. We denote by $C_{0}(\mathbb{R})$ the set of continuous map on $\mathbb{R}$ with zero limits in $\pm\infty$ . For $f\in C_{0}(\mathbb{R})$ and $n\in\mathbb{Z}$ we let $f_{n}=f(\cdot+n)$ be the translation of $f$ by $n$ .

Theorem 6.3.

[AO96] There exists $g\in C_{0}(\mathbb{R})$ such that the vector space spanned by $g_{n}$ , $n\in\mathbb{N}$ is dense in $C_{0}(\mathbb{R})$ .

In particular the operator $V:C_{0}(\mathbb{R})\circlearrowleft$ , $f\mapsto f(\cdot+1)$ , is cyclic. A Borel set $S$ of $\mathbb{R}$ is called a set of uniqueness if the sets $S_{n}:=(S+2\pi n)\cap[-\pi,\pi]$ , $n\in\mathbb{Z}$ satisfy the following properties:

(1)

$S_{n}$ , $n\in\mathbb{Z}$ , are pairwise disjoint,
(2)

$\mathrm{Leb}(S_{n}\cap U)>0$ for any $n\in\mathbb{Z}$ and any open set $U$ of $[-\pi,\pi]$ ,
(3)

$\mathrm{Leb}(S)<\infty$ ,

where $\mathrm{Leb}$ denotes the Lebesgue measure on $\mathbb{R}$ .

Atzmon and Olevskii proved for any set of uniqueness $S$ (such sets exist!) the conclusion of Theorem 6.3 holds true with $g$ being the the Fourier transform of the indicator function of $S$ . Let us just remark that if $S$ is a set of uniqueness then

S^{l}=\bigcup_{n\in\mathbb{Z}}(S_{nk+l}+2\pi k),1\leq l\leq k,

are $k$ disjoints sets of uniqueness. Let $C_{0}(\mathbb{R};\mathbb{C})$ be the set of continuous map on $\mathbb{C}$ with zero limits in infinity.

Lemma 6.4.

The operator $V:C_{0}(\mathbb{R};\mathbb{C})^{k}\circlearrowleft$ , $(f_{i})_{1\leq i\leq k}\mapsto\left(f_{i}(\cdot+1)\right)_{1\leq i\leq k}$ is cyclic. In particular, the operator $U:C_{0}(\mathbb{R})^{k}\circlearrowleft$ , $(f_{i})_{1\leq i\leq k}\mapsto\left(f_{i}(\cdot+1)\right)_{1\leq i\leq k}$ is cyclic.

Proof.

Let $S$ , $S^{l}$ , $1\leq l\leq k$ , be sets of uniqueness as above. By following [AO96] we show that the vector space generated by the translates of $g:=(\widehat{\chi_{S^{l}}})_{1\leq l\leq k}$ is dense in $C_{0}(\mathbb{R};\mathbb{C})^{k}$ with $\widehat{\chi_{S^{l}}}$ be the Fourier transform of the indicator function $\chi_{S^{l}}$ of $S^{l}$ . It follows that the translates of $Re(g)$ is dense in $C_{0}(\mathbb{R})^{k}$ . Let $\mu=(\mu_{l})_{1\leq l\leq k}$ be a complex bounded measure with

\langle V^{n}(g),\mu\rangle=\sum_{1\leq l\leq k}\int(\widehat{\chi_{S^{l}}})_{n}d\mu_{l}=0,

for all $n\in\mathbb{Z}$ . It is enough to prove $\mu_{l}=0$ for all $1\leq l\leq k$ . By Plancherel-Parseval formula we have

\int(\widehat{\chi_{S^{l}}})_{n}d\mu_{l}=\int\widehat{(\widehat{\chi_{S^{l}}})_{n}}(t)\hat{\mu_{l}}(-t)\,dt=-\int\chi_{S^{l}}(t)e^{-int}\hat{\mu_{l}}(t)\,dt.

Therefore we have

\sum_{1\leq l\leq k}\int\chi_{S^{l}}(t)\hat{\mu_{l}}(t)e^{-int}\,dt=0,

for all $n$ . But this term is just the $n^{th}$ coefficient of the function of $L^{1}([-\pi,\pi])$ given by $\sum_{m\in\mathbb{Z}}(\sum_{1\leq l\leq k}\chi_{S^{l}}\hat{\mu}_{l})(\cdot+2\pi m)$ , which should therefore be $0$ . As the sets $(S^{l}+2\pi m)\cap[-\pi,\pi],m\in\mathbb{Z},$ are pairwise disjoint, each term of the previous sum should be zero ; that is $(\chi_{S^{l}}\hat{\mu}_{l})(x+2\pi k)=0$ for all $m,l$ and for Lebesgue almost every $x\in[-\pi,\pi]$ . By Property (2) in the definition of a set of uniqueness, we conclude $\widehat{\mu_{l}}=0$ . Therefore $\mu_{l}=0$ for each $1\leq l\leq k$ and consequently the translates of $g$ is dense in $C_{0}(\mathbb{R};\mathbb{C})^{k}$ . ∎

Proposition 6.5.

Let $f:[0,1]\circlearrowleft$ be a homeomorphism of the interval. Then

\text{\rm Mult}(f)=\sharp\mathcal{M}_{e}([0,1],f).

Proof.

We first deal with the case of an increasing homeomorphism. The ergodic measures of $f$ are the Dirac measures at these fixed points. Notice that $f$ has at least two fixed points, $0$ and $1$ . If it has infinitely many fixed points, then $\text{\rm Mult}(U_{f})\geq\sharp\mathcal{M}_{e}([0,1],f)=\infty$ . Now assume it has finitely many fixed points. Let $2\leq k+1<+\infty$ be the number of fixed points. Since $\varphi(x)-\varphi\circ f(x)=0$ for any continuous function $\varphi\in C(X)$ and any fixed point $x$ , the space $\overline{B_{f}([0,1])}$ is the set of real continuous maps on the interval which vanishes at the fixed points. It follows that the operator $\underline{U}_{f}$ is spectrally conjugate to $V:C_{0}(\mathbb{R})^{k}\circlearrowleft$ , $(f_{i})_{1\leq i\leq k}\mapsto\left(f_{i}(\cdot+1)\right)_{1\leq i\leq k}$ . By Lemma 6.4 we have $\text{\rm Mult}(\underline{U_{f}})=1$ . It follows then from Proposition 2.9 and Proposition 2.6 that

\cdot

\sharp\mathcal{M}_{e}([0,1],f)\leq\text{\rm Mult}(U_{f})\leq\sharp\mathcal{M}_{e}([0,1],f)+\text{\rm Mult}(\underline{U_{f}})-1=\sharp\mathcal{M}_{e}([0,1],f).

It remains to consider the case of a decreasing homeomorphism $f$ . Let $0<a<1$ be the unique fixed point of $f$ . Then $f^{2}:[0,a]\circlearrowleft$ is an increasing homeomorphism. Let $0=x_{1}<x_{2}<\cdots<x_{k}=a$ be the fixed points of $f^{2}|_{[0,a]}$ . Then the ergodic measures of $f$ are the atomic periodic measures $\delta_{a}$ and $\frac{1}{2}\left(\delta_{x_{i}}+\delta_{f(x_{i})}\right)$ for $i=1,\cdots,k-1$ . In particular we have $k=\sharp\mathcal{M}_{e}([0,1],f)$ . From the previous case there is a generating family $\mathcal{G}=\{g_{1},\cdots,g_{k}\}$ for $f^{2}:[0,a]\circlearrowleft$ . Let $h\in C([0,1])$ . For any $\epsilon>0$ , there are $N\in\mathbb{N}$ , $a_{l,n}$ and $b_{l,n}$ , for $l=1,\cdots,k$ and $|n|\leq N$ , (depending on $\epsilon$ ), such that

\cdot

\displaystyle\|h-\sum_{l,n}a_{l,n}g_{l}\circ f^{2n}\|_{{[0,a]},\infty}

\displaystyle<\epsilon,

and

\cdot

\displaystyle\|h\circ f^{-1}-\sum_{l,n}b_{l,n}g_{l}\circ f^{2n}-h(a)\|_{{[0,a]},\infty}

\displaystyle<\epsilon,

where $\|g\|_{{[0,a]},\infty}=\sup_{x\in[0,a]}|g(x)|$ . We consider the extension $\tilde{g_{l}}$ of $g_{l}$ to $[0,1]$ with $\tilde{g_{l}}=g_{l}(a)$ on $[a,1]$ . We check now that $\tilde{\mathcal{G}}=\{\tilde{g}_{1},\cdots,\tilde{g}_{k}\}$ is generating for $f$ . It follows from (6 $\cdot$ 2) and (6 $\cdot$ 3) at $x=a$ that

\cdot

\left|h(a)-\sum_{l,n}a_{l,n}g_{l}(a)\right|<\epsilon\text{ and }\left|\sum_{l,n}b_{l,n}g_{l}(a)\right|<\epsilon.

Observe that

\displaystyle h

\displaystyle=h|_{[0,a]}+(h\circ f^{-1}|_{[0,a]})\circ f|_{[a,1]}.

Combining (6 $\cdot$ 4) with (6 $\cdot$ 2), we obtain that

		$\displaystyle\\|h-\sum_{l,n}a_{l,n}\tilde{g_{l}}\circ f^{2n}-\sum_{l,n}b_{l,n}\tilde{g_{l}}\circ f^{2n+1}\\|_{[0,a],\infty}$
	$\displaystyle=$	$\displaystyle\\|h-\sum_{l,n}a_{l,n}{g_{l}}\circ f^{2n}-\sum_{l,n}b_{l,n}{g_{l}}(a)\\|_{[0,a]\infty}<2\epsilon.$

Similarly, combining (6 $\cdot$ 4) with (6 $\cdot$ 3) we get

		$\displaystyle\\|h-\sum_{l,n}a_{l,n}\tilde{g_{l}}\circ f^{2n}-\sum_{l,n}b_{l,n}\tilde{g_{l}}\circ f^{2n+1}\\|_{[a,1],\infty}$
	$\displaystyle=$	$\displaystyle\\|h-\sum_{l,n}a_{l,n}{g_{l}}(a)-\sum_{l,n}b_{l,n}{g_{l}}\circ f^{2n+1}\\|_{[a,1]\infty}$
	$\displaystyle\leq$	$\displaystyle\|h(a)-\sum_{l,n}a_{l,n}{g_{l}}(a)\|+\\|h-\sum_{l,n}b_{l,n}{g_{l}}\circ f^{2n+1}-h(a)\\|_{[a,1]\infty}<2\epsilon.$

Therefore we have

\|h-\sum_{l,n}a_{l,n}\tilde{g_{l}}\circ f^{2n}-\sum_{l,n}b_{l,n}\tilde{g_{l}}\circ f^{2n+1}\|_{\infty}<2\epsilon.

We conclude that $\tilde{\mathcal{G}}$ is generating for $f$ as $\epsilon>0$ and $h\in C([0,1])$ have been chosen arbitrarily.

∎

Question 6.6.

What is the topological multiplicity of a Morse-Smale diffeomorphism?

6.4. Substitution Subshift

For a finite alphabet $\mathcal{A}$ with $\sharp\mathcal{A}\geq 2$ we let $\mathcal{A}^{+}$ be the set of associated finite words, i.e. finite sequences of letters in $\mathcal{A}$ . For a word $w$ in $\mathcal{A}^{+}$ we let $|w|$ be the number of letters of $w$ . A substitution is a map $\zeta:\mathcal{A}\rightarrow\mathcal{A}^{+}$ . We say $\zeta$ has the constant length if $|\zeta(a)|$ does not depend on $a\in\mathcal{A}$ . The substitution may be extended to finite and infinite (one-sided) words by concatenation: $\zeta(u_{0}u_{1}\cdots)=\zeta(u_{0})\zeta(u_{1})\cdots$ for $u_{i}\in\mathcal{A}$ and $i\in\mathbb{N}$ . In this way $\zeta$ can be iterated.

The langage $\mathcal{L}_{\zeta}$ of $\zeta$ is the set of all subwords of $\zeta^{n}(a)$ over $a\in\mathcal{A}$ and $n\in\mathbb{N}$ , i.e. the set of finite sequence of consecutive letters appearing in the words $\zeta^{n}(a)$ . The associated subshift $X_{\zeta}\subset\mathcal{A}^{\mathbb{Z}}$ is the set of sequences $(x_{n})_{n\in\mathbb{Z}}$ such that $x_{p}x_{p+1}\cdots x_{q}$ belongs to $\mathcal{L}_{\zeta}$ for any $p\leq q\in\mathbb{Z}$ . The map $\zeta$ may be extended to map from $X_{\zeta}$ to $X_{\zeta}$ by letting

\zeta(x)=\cdots\zeta(x_{-1})\zeta(x_{-1})\,|\,\zeta(x_{0})\cdots\zeta(x_{1})\cdots

for any $x=(x_{n})_{n\in\mathbb{Z}}\in X_{\zeta}$ , where the non negative coordinates of $\zeta(x)$ lie after the symbol “ $|$ ”. For $w\in\mathcal{A}$ we recall that the associated cylinder $[w]$ in $X_{\zeta}$ is given by the sequences in $X_{\zeta}$ whose $|w|$ first coordinates coincide with $w$ .

A substitution $\zeta$ is said to be primitive when there exists $n\in\mathbb{N}^{*}$ such that for any letters $a,b$ the word $\zeta^{n}(a)$ contains the letter $b$ . It is well-known that the shift $\sigma$ on $X_{\zeta}$ with primitive $\zeta$ is then minimal and uniquely ergodic (see [Que10]). In the followings, we only consider aperiodic primitive substitutions, i.e. primitive substitutions such that $X_{\zeta}$ is not reduced to a periodic orbit.

We will make use of the standard recognazibility property of primitive substitutions:

Theorem 6.7.

[Mos96] Let $\zeta$ be an aperiodic primitive substitution. Then for any positive integer $n$ , the family

P_{n}:=\{\sigma^{k}\left(\zeta^{n}[a]\right),\ a\in\mathcal{A}\ \text{ and }0\leq k<|\zeta^{n}(a)|\}

defines a partition of $X_{\zeta}$ into clopen sets.

Moreover $\zeta^{n}:X_{\zeta}\rightarrow\zeta^{n}(X_{\zeta})$ is a topological conjugacy between $(X_{\zeta},\sigma)$ and the induced system $\left(\zeta^{n}(X_{\zeta}),\sigma_{\zeta^{n}(X_{\zeta})}\right)$ defined as $\sigma_{\zeta^{n}(X_{\zeta})}(x)=\sigma^{|\zeta^{n}(x_{0})|}(x)$ for any $x=(x_{k})_{k\in\mathbb{Z}}\in\zeta^{n}(X_{\zeta})$ .

We define two equivalence relations on $\mathcal{A}$ as follows:

a\sim^{d}b\text{ if and only if }\text{ $\zeta(a)$ and $\zeta(b)$ begin with the same letter};

a\sim^{f}b\text{ if and only if }\text{ $\zeta(a)$ and $\zeta(b)$ end with the same letter}.

We then let $\mathcal{A}_{d}$ and $\mathcal{A}_{f}$ be respectively the equivalence classes of $\sim^{d}$ and $\sim^{f}$ . We let $N_{d}$ and $N_{f}$ be the number of equivalence classes for $\sim^{d}$ and $\sim^{f}$ .

Theorem 6.8.

Let $\zeta$ be a primitive aperiodic substitution (resp. with constant length). Then the multiplicity $(X_{\zeta},\sigma)$ is less than or equal to (resp. strictly less than)

K_{\zeta}=\sharp\mathcal{A}(N_{d}+N_{f})+\min(N_{d},N_{f})-(\sharp\mathcal{A}+N_{d}N_{f}),

which is less than or equal to $\sharp\mathcal{A}^{2}$ .

For a proper substitution (i.e. when $N_{d}=N_{f}=1$ ) we get $K_{\zeta}=\sharp\mathcal{A}$ . The partitions $(P_{n})_{n\in\mathbb{N}}$ in Theorem 6.7 are not refining, i.e. the diameter of $P_{n}$ does not go to zero with $n$ in general. To prove Theorem 6.8 we use the following finer clopen partitions $(Q_{n})_{n\in\mathbb{N}}$ . For a subset $J\subset\mathcal{A}$ and $a\in\mathcal{A}$ we let $[a|J]\subset X_{\zeta}$ (resp. $[J|a]$ ) be the union of $\sigma([aj])$ (resp. $\sigma([ja])$ ) over $j\in J$ ).

Lemma 6.9.

Let $\zeta$ be an aperiodic primitive substitution and let $M_{n}$ be the integer part of $\min_{a\in\mathcal{A}}|\zeta^{n-1}(a)|/2$ . Then for any positive integer $n$ , the family

	$\displaystyle Q_{n}:=$	$\displaystyle\{\sigma^{l}\left(\zeta^{n}[J_{f}\|a]\right):\ a\in\mathcal{A},\ J_{f}\in\mathcal{A}_{f}\text{ and }0\leq l<M_{n}\}\bigcup$
		$\displaystyle\{\sigma^{-l}\left(\zeta^{n}[a\|J_{d}]\right):\ a\in\mathcal{A},\ J_{d}\in\mathcal{A}_{d}\text{ and }0\leq l<\|\zeta^{n}(a)\|-M_{n}\}$

defines a partition of $X_{\zeta}$ into clopen sets whose diameter is going to zero when $n$ goes to infinity.

Proof.

Clearly, for $0\leq k<M_{n}$ , the clopen sets $\sigma^{k}\left(\zeta^{n}[J_{f}|a]\right)$ for $J_{f}\in\mathcal{A}_{f}$ defines a partition of $\sigma^{k}(\zeta^{n}[a])$ . On the other hand, for $0\leq l<|\zeta^{n}(a)|-M_{n}$ , the clopen sets $\sigma^{-l}\left(\zeta^{n}[a|J_{d}]\right)$ for $J_{d}\in\mathcal{A}_{d}$ defines a partition of $\sigma^{|\zeta^{n}(a)|-l}(\zeta^{n}[a])$ . Therefore we deduce then from Theorem 6.7 that $Q_{n}$ is a partition of $X_{\zeta}$ . It remains to show that their diameter is going to zero when $n$ goes to infinity. Fix $a\in\mathcal{A}$ and $J_{f}\in\mathcal{A}_{f}$ and let $b\in\mathcal{A}$ such that $\zeta(j)$ ends with the letter $b$ for any $j\in J_{f}$ . Then any sequence in $\zeta^{n}[J_{f}|a]$ lies in $\sigma^{|\zeta^{n-1}(b)|}\left([\zeta^{n-1}(b)\zeta^{n}(a)]\right)$ . In particular, for any $x,y\in\sigma^{l}\left(\zeta^{n}[J_{f}|a]\right)$ with $0\leq l<M_{n}$ , we have $x_{-M_{n}+1}\cdots x_{M_{n}-1}=y_{-M_{n}+1}\cdots y_{M_{n}-1}$ since we have $M_{n}\leq\max(|\zeta^{n}(a)|/2,|\zeta^{n-1}(b)|)$ . Arguing similarly with $\sigma^{-l}\left(\zeta^{n}[a|J_{d}]\right)$ we conclude that the diameter of $Q_{n}$ goes to zero when $n$ goes to infinity, because $M_{n}$ goes to infinity. ∎

For a finite collection $\mathcal{C}$ of clopen sets and for $n\in\mathbb{N}^{*}$ we let $\mathcal{F}_{n}^{\mathcal{C}}$ be the set of indicator functions $\chi_{\zeta^{n}(C)}$ with $C\in\mathcal{C}$ . We also write $\mathcal{F}(Q_{n})$ for the set of indicator functions $\chi_{A}$ with $A\in Q_{n}$ .

Lemma 6.10.

There is a finite collection $\mathcal{C}$ of clopen sets with $\sharp\mathcal{C}\leq K_{\zeta}$ , such that

\forall n\in\mathbb{N}^{*},\ V_{\mathcal{F}(Q_{n})}=V_{\mathcal{F}_{n}^{\mathcal{C}}}.

Proof.

Without loss of generality we may assume $N_{d}\leq N_{f}$ . Fix $\overline{J_{f}}\in\mathcal{A}_{f}$ and $\overline{i_{d}}\in J_{d}$ for any $J_{d}\in\mathcal{A}_{d}$ . We consider the collection

\mathcal{C}:=\left\{[a|J_{d}],\ [J_{f}|b]\ :\ a\in\mathcal{A},\ J_{d}\in\mathcal{A}_{d},\ J_{f}\in\mathcal{A}_{f}\setminus\{\overline{J_{f}}\},\ b\in J_{d}\setminus\{\overline{i_{d}}\}\ \right\}.

Clearly $V_{\mathcal{F}(Q_{n})}\supset V_{\mathcal{F}_{n}^{\mathcal{C}}}$ and

\sharp\mathcal{C}\leq N_{d}\sharp\mathcal{A}+(N_{f}-1)(\sharp\mathcal{A}-N_{d})=K_{\zeta}.

We show now the other inclusion $V_{\mathcal{F}(Q_{n})}\subset V_{\mathcal{F}_{n}^{\mathcal{C}}}$ . Let $\tilde{\mathcal{C}}_{n}$ be the collection of clopen sets $E$ satisfying $\chi_{\zeta^{n}(E)}\in V_{\mathcal{F}_{n}^{\mathcal{C}}}$ . It is enough to check $[a|J_{d}]$ and $[J_{f}|b]$ lie in $\tilde{\mathcal{C}}_{n}$ for any $a,b\in\mathcal{A}$ , $J_{d}\in\mathcal{A}_{d}$ and $J_{f}\in\mathcal{A}_{f}$ . For $E,E^{\prime}\in\tilde{\mathcal{C}}_{n}$ , the union $E\cup E^{\prime}$ (resp. $E\setminus E^{\prime}$ ) also belongs to $\tilde{\mathcal{C}}_{n}$ if $E\cap E^{\prime}=\emptyset$ (resp. $E^{\prime}\subset E$ ), because $\chi_{\zeta^{n}(E\cup E^{\prime})}=\chi_{\zeta^{n}(E^{\prime})}+\chi_{\zeta^{n}(E)}$ (resp. $\chi_{\zeta^{n}(E\setminus E^{\prime})}=\chi_{\zeta^{n}(E)}-\chi_{\zeta^{n}(E^{\prime})}$ ) as $\zeta^{n}$ is one-to-one.

Then for any $J_{d}\in\mathcal{A}_{d}$ and any $J_{f}\in\mathcal{A}_{f}$ , the set $[J_{f}|J_{d}]=\coprod_{a\in J_{f}}[a|J_{d}]$ lie in $\tilde{\mathcal{C}}_{n}$ , because they are disjoint unions of elements of $\mathcal{C}\subset\tilde{\mathcal{C}}_{n}$ . We have also $[J_{f}|\overline{i_{d}}]=[J_{f}|J_{d}]\setminus\left(\coprod_{J_{d}\ni b\neq\overline{i_{d}}}[J_{f}|b]\right)\in\tilde{\mathcal{C}}_{n}$ for any $J_{f}\neq\overline{J_{f}}$ . Finally for any $b\in\mathcal{A}$ we get $[\overline{J_{f}}|b]=[b]\setminus\left(\coprod_{J_{f}\neq\overline{J_{f}}}[J_{f}|b]\right)$ . But $[b]=\sigma^{-1}([b|])$ , then $\zeta^{n}([b])=\sigma^{-|\zeta^{n}(b)|}\left(\zeta^{n}([b|])\right)$ and $[b|]=\coprod_{J_{d}}[b|J_{d}]\in\tilde{\mathcal{C}}_{n}$ . Therefore $\chi_{\zeta^{n}([b])}=\chi_{\zeta^{n}([b|])}\circ\sigma^{|\zeta^{n}(b)|}\in V_{\mathcal{F}_{n}^{\mathcal{C}}}$ , thus $[b]\in\tilde{\mathcal{C}}_{n}$ . We conclude that $[\overline{J_{f}}|b]$ also belongs to $\tilde{\mathcal{C}}_{n}$ .

∎

Proof of Theorem 6.9.

Let $\mathcal{C}$ be the collection given by Lemma 6.10. Then for any $A\in Q_{n}$ , the function $\chi_{A}$ belongs to $V_{\mathcal{F}_{n}^{\mathcal{C}}}$ . As the diameter of $Q_{n}$ goes to zero with $n$ , the hypothesis of Lemma 4.1 are satisfied, so that $\text{\rm Mult}(X_{\zeta},\sigma)\leq\sharp\mathcal{C}\leq K_{\zeta}$ .

When moreover the substitution has constant lenght $L$ , then the induced system on $\zeta^{n}(X_{\zeta})$ is just the power $\sigma^{L^{n}}$ . By Theorem 6.7 the induced system is topologically conjugated to the substitution system, thus it is also uniquely ergodic. Let $\nu_{n}$ be the corresponding unique invariant measure. In particular for any clopen subset $C$ of $\zeta^{n}(X)$ , the sequence of continuous functions

\frac{1}{K}\sum_{0\leq k\leq K}\chi_{C}\circ\sigma^{L^{n}}

is converging uniformly to $\nu_{n}(C)\chi_{\zeta^{n}(X)}$ when $K$ goes to infinity. When defining $\mathcal{C}$ in the previous lemma, we let all $[a|J_{d}]$ in $\mathcal{C}$ for $a\in\mathcal{A}$ and $J_{d}\in\mathcal{A}_{d}$ . The elements of this form defines a partition and we may choose one such element $[\overline{a}|\overline{J_{d}}]$ such that $\zeta^{n}([\overline{a}|\overline{J_{d}}])$ has not full measure for the induced system. We can remove this element from $\mathcal{C}$ . We let $\mathcal{C}^{\prime}$ be this new collection. If $C^{\prime}=[a|J_{d}]$ is an element of $\mathcal{C}^{\prime}$ with $\nu_{n}(\zeta^{n}(C^{\prime}))>0$ , then $\chi_{\zeta^{n}(X)}$ belongs to $V_{\chi_{\zeta^{n}(C^{\prime})}}$ therefore $\chi_{\zeta^{n}([\overline{a}|\overline{J_{d}}])}$ lies in $V_{\mathcal{F}_{n}^{\mathcal{C}^{\prime}}}$ , i.e. $V_{\mathcal{F}_{n}^{\mathcal{C}}}=V_{\mathcal{F}_{n}^{\mathcal{C}^{\prime}}}$ . Therefore $\text{\rm Mult}(X_{\zeta},\sigma)\leq\sharp\mathcal{C}^{\prime}=\sharp\mathcal{C}-1<K_{\zeta}$ .

∎

For the Thue-Morse substitution: $0\mapsto 01$ and $1\mapsto 10$ , we get $N_{d}=N_{f}=2$ , thus $\text{\rm Mult}(X_{\zeta},\sigma)<K_{\zeta}=4$ .

Question 6.11.

What is the topological multiplicity of the Thue-Morse substitution?

7. Subshifts with linear growth complexity

We consider a subshift $X\subset\mathcal{A}^{\mathbb{Z}}$ with letters in a finite alphabet $\mathcal{A}$ . For $x\in\mathcal{A}^{\mathbb{Z}}$ we denote by $x=(x_{n})_{n\in\mathbb{Z}}$ for $x_{n}\in\mathcal{A}$ . Let $\mathcal{L}_{n}(X)\subset\mathcal{A}^{n}$ be the finite words of $X$ of length $n$ , i.e. $\mathcal{L}_{n}(X)=\{x_{k}x_{k+1}\dots x_{k+n-1}:x\in X,k\in\mathbb{Z}\}$ . The word complexity of $X$ is given by

\forall n\in\mathbb{N},\ \ p_{X}(n)=\sharp\mathcal{L}_{n}(X).

We suppose that $X$ is aperiodic and has linear growth, that is, for some $k\in\mathbb{N}^{*}$

\cdot

\liminf_{n}\frac{p_{X}(n)}{n}\leq k.

Boshernitzan [Bos92] showed that such a subshift admits at most $k$ ergodic measures. By Theorem 5.5 in [DDMP21] such subshifts, when assumed to be moreover minimal, have topological rank less than or equal to $(1+k\sharp\mathcal{A}^{2})^{2(k+2)}$ . We show in this section the following upper bound on the topological multiplicity.

Theorem 7.1.

Any aperiodic subshift $X$ with $\liminf_{n\to\infty}\frac{p_{X}(n)}{n}\leq k$ has topological multiplicity less than or equal to $2k$ .

One may wonder if the upper bound in Theorem 7.1 is sharp.

Question 7.2.

Is an aperiodic subshift $X$ with $\liminf_{n\to\infty}\frac{p_{X}(n)}{n}=1$ topologically simple?

In order to prove Theorem 7.1, we define some notations. Let $Q_{n}$ be the subset of $\mathcal{L}_{n}(X)$ given by words $w$ such that there are several letters $a\in\mathcal{A}$ with $wa\in\mathcal{L}_{n+1}(X)$ . We also let $Q^{\prime}_{n+1}$ be the $(n+1)$ -words $wa$ as above. Clearly, we have

\cdot

\sharp Q_{n}\leq p_{X}(n+1)-p_{X}(n)\text{ and }\sharp Q^{\prime}_{n+1}=\sharp Q_{n}+p_{X}(n+1)-p_{X}(n).

Through this section, we always assume the subshift is aperiodic and satisfies the linear growth (7 $\cdot$ 1).

Lemma 7.3.

The subset of integers

\mathcal{N}=\{n\in\mathbb{N}:p_{X}(n+1)<(k+1)(n+1)\text{ and }p_{X}(n+1)-p_{X}(n)\leq k\}.

is infinite.

For the sake of completeness, we reproduce the proof, which is contained in Theorem 2.2 of [Bos84].

Proof.

By (7 $\cdot$ 1), we have

\liminf_{n}(p_{X}(n)-(k+1)n)=-\infty.

It follows that

\mathcal{M}=\left\{n\in\mathbb{N}:p_{X}(n+1)-(k+1)(n+1)\leq\min\{0,\min_{1\leq m\leq n}\{p_{X}(m)-(k+1)m\}\}-1\right\}

is an infinite set. For any $n\in\mathcal{M}$ , we have

p_{X}(n+1)-p_{X}(n)\leq(k+1)(n+1)-(k+1)n-1=k.

On the other hand, for any $n\in\mathcal{M}$ , we get

p_{X}(n+1)\leq(k+1)(n+1)-1.

This implies that $\mathcal{M}\subset\mathcal{N}$ . Therefore, the set $\mathcal{N}$ is infinite. ∎

Lemma 7.4 ([Bos84], Lemma 4.1).

For any $n\in\mathcal{N}$ and $m\geq(k+2)(n+1)$ , any word $w\in\mathcal{L}_{m}$ contains a subword in $Q_{n}$ .

For the sake of completeness we provide a proof here.

Proof.

We prove it by contradiction. Assume to the contrary that all $(m-n+1)$ $n$ -subwords of $w$ do not belong to $Q_{n}$ . That means that each of these $n$ -blocks determines uniquely the next letter. Since $m-n+1\geq 2(k+1)n>p_{X}(n)$ , at least one $n$ -word appears more than one time as a subword of $w$ . Therefore $X$ contains a periodic point. This contradicts our assumption. ∎

Now we show that any cylinder of length less than $n$ can be decomposed as the cylinders of elements in $Q^{\prime}_{n+1}$ after translations.

Lemma 7.5.

Let $n\in\mathcal{N}$ . Any cylinder $[w]$ with length of $w$ less than $n$ may be written uniquely as a finite disjoint union of sets of the form $\sigma^{p}[q^{\prime}_{n+1}]$ with $p\in\mathbb{N}$ , $q^{\prime}_{n+1}\in Q^{\prime}_{n+1}$ , such that $\sigma^{t}[q^{\prime}_{n+1}]\cap[q_{n}]=\emptyset$ for any $0<t<p$ and any $q_{n}\in Q_{n}$ .

Remark that by Lemma 7.4 the integers $p$ belongs to $[0,(k+2)(n+1)]$ .

Proof.

Let $[w]$ be a cylinder associated to a word $w\in\mathcal{L}_{l}(X)$ with $l<n$ . For $x\in[w]$ , we let $K_{x}\in\mathbb{Z}$ be the largest integer $j$ less than $l$ such that $x_{j-n+1}\cdots x_{j}$ belongs to $Q_{n}$ . Then the word $w_{n+1}^{x}=x_{K_{x}-n+1}\cdots x_{K_{x}+1}$ belongs to $Q^{\prime}_{n+1}$ . Observe also that by Lemma 7.4 we have $l-1-K_{x}\leq(k+2)(n+1)$ , thus $n-1-K_{x}\leq(k+3)(n+1)$ . Let $W_{n+1}$ be the collection of these words $w_{n+1}^{x}$ over $x\in[w]$ . By definition of $K_{x}$ and $Q_{n}$ the word $w_{n+1}^{x}$ completely determines the $l-1-K_{x}$ next letters, that is to say,

[w_{n+1}^{x}]=[x_{K_{x}-n+1}\cdots x_{l-1}].

For $x,y\in[w]$ the sets $\sigma^{n-1-K_{x}}[w_{n+1}^{x}]$ and $\sigma^{n-1-K_{y}}[w_{n+1}^{y}]$ are either disjoint or equal. As $x$ belongs to $[w]$ we have $x\in\sigma^{n-1-K_{x}}[w_{n+1}^{x}]\subset[w]$ and finally

[w]=\coprod_{w_{n+1}^{x}\in W_{n+1}}\sigma^{n-1-K_{x}}w^{x}_{n+1}.

We complete the proof. ∎

Proof of Theorem 7.1.

By (7 $\cdot$ 2) and the definition of $\mathcal{N}$ we have for $n\in\mathcal{N}$ :

	$\displaystyle\sharp Q^{\prime}_{n+1}$	$\displaystyle=\sharp Q_{n}+p_{X}(n+1)-p_{X}(n),$
		$\displaystyle\leq 2(p_{X}(n+1)-p_{X}(n)),$
		$\displaystyle\leq 2k.$

For $n\in\mathcal{N}$ we let $F_{n}=\{\chi_{[q^{\prime}_{n+1}]},\ q^{\prime}_{n+1}\in Q^{\prime}_{n+1}\}$ . By Lemma 7.5, any cylinder $[w]$ with length less than $n$ is a finite disjoint union of $\sigma^{p}[q^{\prime}_{n+1}]$ . In particular $\chi_{[w]}$ lies in $V_{F_{n}}$ . We may therefore apply Lemma 4.1 to $(F_{n})_{n\in\mathcal{N}}$ and we get

\text{\rm Mult}(X,\sigma)\leq\sup_{n\in\mathcal{N}}\sharp Q^{\prime}_{n+1}\leq 2k.

∎

7.1. Multiplicity of invariant measures

It follows from Theorem 7.1 and Lemma 2.4 that any ergodic measure has (ergodic) multiplicity bounded by $2k$ . In fact we may refine this result as follows:

Theorem 7.6.

Let $X$ be an aperiodic subshift with $\liminf_{n\to\infty}\frac{p_{X}(n)}{n}\leq k$ . Then

\sum_{\mu\in\mathcal{M}_{e}(X,\sigma)}\text{\rm Mult}(\mu)\leq 2k.

In order to prove Theorem 7.6, we first recall some notations and then show two lemmas for general aperiodic subshifts. We have learned from the referee that some parts of our proofs overlap with results in [Cre23][Esp23]. Let $(Y,\sigma)$ be an aperiodic subshift. For two finite words $w$ and $v$ , we denote by $N(w|v)$ the number of times that $w$ appears as a subword of $v$ . Also, We define $d(w|v)=N(w|v)/|v|$ . For a generic point $x$ of a measure $\mu$ , we have

\lim_{n\to\infty}d(w|x_{1}^{n})=\mu([w]),

where $x_{1}^{n}=x_{1}x_{2}\dots x_{n}$ . For a finite word $v$ , we denote by $v^{\otimes m}=\underbrace{vv\dots v}_{m\text{ times}}$ . For a finite word $w$ , we denote by

\nu_{w}=\frac{1}{|w|}\sum_{k=0}^{|w|-1}\delta_{\sigma^{k}(\bar{w})},

where $|w|$ is the length of $w$ and $\bar{w}\in\mathcal{A}^{\mathbb{Z}}$ is the periodization of $w$ , i.e. $w^{\otimes\infty}$ .

Let $w_{n}$ be a word of length $n$ . Foy any $n$ , we put $\ell_{n}=\ell(w_{n}):=\min\{1\leq\ell<n:[w_{n}]\cap\sigma^{\ell}([w_{n}])\not=\emptyset\}$ and $L_{n}:=1+\sharp\{1\leq\ell<n:[w_{n}]\cap\sigma^{\ell}([w_{n}])\not=\emptyset\}$ , with the convention $\min\emptyset=n$ . Let $v_{n}=v(w_{n})$ be the first $\ell_{n}$ -subword of $w_{n}$ . It follows that $w_{n}=v_{n}^{\otimes K_{n}}\hat{v}_{n}$ with $\hat{v}_{n}\neq v_{n}$ being a prefix of $v_{n}$ . Then $K_{n}=\lfloor n/\ell_{n}\rfloor\geq L_{n}$ . Observe that for any $x\in X$ , $p\geq\ell_{n}$ and any word $u$ of length less than $\ell_{n}$ we have

(7 $\cdot$ 3)	$\displaystyle N(u\|x_{1}^{p})$	$\displaystyle\geq N(u\|w_{n})\frac{N(w_{n}\|x_{1}^{p})}{L_{n}},$
	$\displaystyle\geq N(u\|v_{n})K_{n}\frac{N(w_{n}\|x_{1}^{p})}{L_{n}},$
	$\displaystyle\geq N(u\|v_{n})N(w_{n}\|x_{1}^{p}).$

In the next two lemmas we assume that

•

the subshift $(Y,\sigma)$ is aperiodic;
•

$w_{n}\in\mathcal{L}_{n}(Y)$ for $n\in\mathbb{N}$ ;
•

$\nu_{w_{n}}\xrightarrow[n\in\mathcal{N}]{n\to+\infty}\nu\in\mathcal{M}_{e}(Y,\sigma)$ , i.e. $\nu_{w_{n}}$ is weakly converging to an ergodic measure $\nu$ when $n$ goes to infinity along a subsequence $\mathcal{N}$ .

Lemma 7.7.

Under the above assumption, we have

\ell_{n}=|v_{n}|\xrightarrow[n\in\mathcal{N}]{n\to+\infty}+\infty

and

\nu_{v_{n}}\xrightarrow[n\in\mathcal{N}]{n\to+\infty}\nu.

Proof.

We argue by contradiction. Assume $(\ell_{n})_{n\in\mathbb{N}}$ has a bounded infinite subsequence $\mathcal{N}^{\prime}$ of $\mathcal{N}$ . Then there are finite words $v$ and $\hat{v}_{n}$ with $|\hat{v}_{n}|<|v|$ such that $w_{n}=v^{\otimes K_{n}}\hat{v}_{n}$ for $n\in\mathcal{N}^{\prime\prime}$ where $\mathcal{N}^{\prime\prime}$ is some infinite subsequence of $\mathcal{N}^{\prime}$ . Observe firstly that the length of $w_{n}$ goes to infinity. As a consequence, $K_{n}$ goes also to infinity as $n$ goes to infinite along $\mathcal{N}^{\prime\prime}$ . But then $X$ should contain the periodic point $\overline{v}$ associated to $v$ which is a contradiction to the aperiodicity of $(Y,\sigma)$ . Therefore $\ell_{n}\xrightarrow[n\in\mathcal{N}]{n\to+\infty}+\infty$ .

Let us check now that $\nu_{v_{n}}\xrightarrow[n\in\mathcal{N}]{n\to+\infty}\nu$ . Let $\nu^{\prime}=\lim_{k\to\infty}\nu_{v_{n_{k}}}$ be a weak limit of $(\nu_{v_{n}})_{n\in\mathcal{N}}$ with a subsequence $(n_{k})_{k\in\mathbb{N}}$ of $\mathcal{N}$ . For any word $u$ with $|u|<\ell_{n}$ , by (7 $\cdot$ 3) we have

N(u|w_{n})\geq N(u|v_{n})K_{n},

and consequently

	$\displaystyle d(u\|w_{n})\geq$	$\displaystyle d(u\|v_{n})\frac{K_{n}\|v_{n}\|}{(K_{n}+1)\|v_{n}\|},$
	$\displaystyle\geq$	$\displaystyle\frac{1}{2}d(u\|v_{n}).$

For any cylinder $[u]$ , By letting $n$ got infinity we get that

\nu([u])\geq\frac{1}{2}\nu^{\prime}([u]).

It implies that $\nu-\frac{1}{2}\nu^{\prime}$ is an $\sigma$ -invariant measure. It follows from the ergodicity of $\nu$ that $\nu=\nu^{\prime}$ . ∎

Lemma 7.8.

For any ergodic measure $\mu\not=\nu$ , we have

\lim_{n\in\mathcal{N},n\to\infty}|v_{n}|\mu([w_{n}])=0.

Proof.

Assume $\limsup_{n\in\mathcal{N},n\to\infty}|v_{n}|\mu([w_{n}])>0.$ By passing to an infinite subsequence $\mathcal{N}^{\prime}$ of $\mathcal{N}$ we have $\lim_{n\in\mathcal{N}^{\prime},n\to\infty}|v_{n}|\mu([w_{n}])=b>0$ . By Lemma 7.7 the sequence $\nu_{v_{n}}$ , $n\in\mathcal{N}^{\prime}$ , is converging to the measure $\nu$ .

Let $x$ be a generic point of $\mu$ . Then we have for any $n$

\cdot

\lim_{p\to\infty}\frac{1}{p}\sum_{\ell=0}^{p-1}\chi_{[w_{n}]}(\sigma^{\ell}(x))=\lim_{p\to\infty}\frac{N(w_{n}|x_{1}^{p})}{p}=\mu([w_{n}]).

In particular for any $n$ we can choose $P_{n}\in\mathbb{N}$ such that for $p\geq P_{n}$

\cdot

\frac{N(w_{n}|x_{1}^{p})}{p}\geq\frac{\mu([w_{n}])}{2}.

Pick an arbitrary cylinder $[u]$ . by Lemma 7.7 there exists an integer $N$ such that for $n>N$ we have

\cdot

N(u|v_{n})\geq\frac{1}{2}\nu([u])|v_{n}|.

It follows from (7 $\cdot$ 3) that

d(u|x^{p}_{1})\geq\frac{1}{4}\nu([u])|v_{n}|\mu(w_{n}).

By letting $p$ , then $n\in\mathcal{N}^{\prime}$ go to infinity, we get for any cylinder $[u]$

\mu([u])\geq\frac{b}{4}\nu([u]).

It implies that $\mu-\frac{b}{4}\nu$ is an $\sigma$ -invariant measure which is a contradiction to the ergodicity of $\mu$ . ∎

We recall now briefly the proof of Boshernitzan that an aperiodic subshift of linear growth has finite many ergodic measures. Let $\mathcal{N}$ be the infinite set as in Lemma 7.3. For any $n\in\mathcal{N}$ , one can choose (not uniquely) an ordered $k$ -tuple of $n$ -words $K_{n}:=\{q_{n,1},\dots,q_{n,k}\}$ which coincides with $Q_{n}$ . By passing to a subsequence $\mathcal{N}^{\prime}$ of $\mathcal{N}$ , we can make each of the sequences of $\nu_{q_{n,i}}$ weakly convergences to some measures $\mu_{i}\in\mathcal{M}(X,T)$ . Boshernitzan showed that

\mathcal{M}_{e}(X,T)\subset\{\mu_{1},\mu_{2},\dots,\mu_{k}\}.

Since $\mu_{i}$ may coincide with the other $\mu_{j}$ for $j\not=i$ , we define $I_{i}=\{1\leq j\leq k:\mu_{j}=\mu_{i}\}$ .

We will use the following complement of Lemma 7.5.

Lemma 7.9.

In the decomposition of a cylinder $[w]$ given by Lemma 7.5, for any term $\sigma^{p}[q^{\prime}_{n+1}]$ with $|v(q^{\prime}_{n+1})|<n+1$ we have $p\leq|v(q^{\prime}_{n+1})|$ .

Proof.

We argue by contradiction. To simplify the notations we write $v_{n}=v(q^{\prime}_{n+1})$ . Assume $|v_{n}|<n+1$ and $p>|v_{n}|$ . By definition of $v_{n}$ we have

\emptyset\not=\sigma^{p}[q^{\prime}_{n+1}]\cap\sigma^{p-|v_{n}|}[q^{\prime}_{n+1}].

But it follows from Lemma 7.5 that $\sigma^{p}[q^{\prime}_{n+1}]$ does not intersect $\sigma^{l}\left(\bigcup_{q_{n}\in Q_{n}}[q_{n}]\right)$ for $0<l<p$ , therefore with $q_{n}\in Q_{n}$ being the prefix of $q^{\prime}_{n+1}$ we get the contradiction

\sigma^{p}[q^{\prime}_{n+1}]\cap\sigma^{p-|v_{n}|}[q^{\prime}_{n+1}]\subset\sigma^{p}[q^{\prime}_{n+1}]\cap\sigma^{p-|v_{n}|}[q_{n}]=\emptyset.

Thus we have $p\leq|v(q^{\prime}_{n+1})|$ . ∎

For a given $i$ we let $(q^{l}_{n,i})_{l\in\mathcal{Q}_{n,i}}$ be the elements of $Q^{\prime}_{n+1}$ with prefix $q_{n,i}$ , where $\mathcal{Q}_{n,i}$ is a subset of $\mathcal{A}$ for each $n\in\mathcal{N}^{\prime},1\leq i\leq k$ . Note that $\nu_{q^{l}_{n,i}}$ is also converging to $\mu_{i}$ for any $l$ when $n$ goes to infinity along $\mathcal{N}^{\prime}$ . Finally we let $v_{n,i}^{l}=v_{n}(q^{l}_{n,i})$ for each $n\in\mathcal{N}^{\prime},1\leq i\leq k$ and $l\in\mathcal{Q}_{n,i}$ .

Proof of Theorem 7.6.

Pick an arbitrary cylinder $[w]$ . Let

[w]=\coprod_{j,l}\coprod_{\in P_{n,j}^{l}}\sigma^{p}[q_{n,j}^{l}],

be the decomposition of $[w]$ given by Lemma 7.5. Recall that $P_{j,l}$ is a subset of $[0,(k+2)(n+1)]$ for any $l$ and by Lemma 7.9 we have also $P_{j,l}\subset[0,|v^{l}_{n,j}|-1]$ if $|v^{l}_{n,j}|<n+1$ . For each $n\in\mathcal{N}^{\prime}$ , we decompose $\{(j,l):1\leq j\leq k,l\in\mathcal{Q}_{n,j}\}$ into three set $J_{n,i}$ , $J_{n,i}^{\prime}$ and $J_{n,i}^{\prime\prime}$ , where $J_{n,i}:=\{(j,l):j\in I_{i}\}$ , $J_{n,i}^{\prime}:=\{(j,l):j\notin I_{i},|v^{l}_{n,j}|=n+1\}$ and $J_{n,i}^{\prime\prime}$ is the rest. Then for $(j,l)\in J_{n,i}^{\prime}$ we have

\cdot

\mu_{i}\left(\coprod_{p\in P_{n,j}^{l}}\sigma^{p}[q_{n,j}^{l}]\right)\leq(k+2)(n+1)\mu_{i}([q_{n,j}^{l}])=(k+2)|v^{l}_{n,j}|\mu_{i}([q_{n,j}^{l}]).

On the other hand, for $(j,l)\in J_{n,i}^{\prime\prime}$ , we have

\cdot

\mu_{i}\left(\coprod_{p\in P_{n,j}^{l}}\sigma^{p}[q_{n,j}^{l}]\right)\leq|v^{l}_{n,j}|\mu_{i}([q_{n,j}^{l}]).

By summing up (7 $\cdot$ 7) and (7 $\cdot$ 8), we have

\cdot

\mu_{i}\left(\coprod_{(j,l)\in J_{n,i}^{\prime}\cup J_{n,i}^{\prime\prime}}\coprod_{p\in P_{n,j}^{l}}\sigma^{p}[q_{n,j}^{l}]\right)\leq 2k(k+2)\sum_{(j,l)\in J_{n,i}^{\prime}\cup J_{n,i}^{\prime\prime}}|v^{l}_{n,j}|\mu_{i}([q_{n,j}^{l}]).

Combining this with Lemma 7.8, we obtain

\lim_{n\to\infty}\mu_{i}\left(\coprod_{(j,l)\in J_{n,i}^{\prime}\cup J_{n,i}^{\prime\prime}}\coprod_{p\in P_{n,j}^{l}}\sigma^{p}[q_{n,j}^{l}]\right)=0.

Therefore we have

	$\displaystyle\left\\|\chi_{[w]}-\sum_{(n,j)\in J_{n,i},\ p\in P_{n,j}^{l}}\chi_{[q_{n,j}^{l}]}\circ\sigma^{-p}\right\\|^{2}_{L^{2}(\mu_{i})}$	$\displaystyle=\left\\|\chi_{[w]}-\chi_{\coprod_{(n,j)\in J_{n,i},\ p\in P_{n,j}^{l}}\sigma^{p}[q_{n,j}^{l}]}\right\\|^{2}_{L^{2}(\mu_{i})},$
		$\displaystyle=\mu_{i}\left(\coprod_{(j,l)\in J_{n,i}^{\prime}\cup J_{n,i}^{\prime\prime}}\coprod_{p\in P_{n,j}^{l}}\sigma^{p}[q_{n,j}^{l}]\right)\xrightarrow[n\in\mathcal{N}^{\prime}]{n\to+\infty}0.$

Thus we can apply Lemma 4.1 in $L^{2}_{0}(\mu_{i})$ with $F_{n}=\{\chi_{[q_{n,j}^{l}]}\ :\ (j,l)\in J_{n,i}\}$ to get

\text{\rm Mult}(\mu_{i})\leq\liminf_{n\in\mathcal{N}^{\prime},n\to\infty}\sharp J_{n,i}.

By summing it up, we conclude that

	$\displaystyle\sum_{\mu\in\mathcal{M}_{e}(X,\sigma)}\text{\rm Mult}(\mu)\leq$	$\displaystyle\sum_{i}\liminf_{n\in\mathcal{N}^{\prime},n\to\infty}\sharp J_{n,i},$
	$\displaystyle\leq$	$\displaystyle\liminf_{n\in\mathcal{N}^{\prime},n\to\infty}\sharp Q^{\prime}_{n+1},$
	$\displaystyle\leq$	$\displaystyle 2k.$

∎

References

[AO96] A. Atzmon and A. Olevskiĭ. Completeness of integer translates in function spaces on ${\bf R}$ . J. Approx. Theory, 87(3):291–327, 1996.
[Bax71] J. R. Baxter. A class of ergodic transformations having simple spectrum. Proc. Amer. Math. Soc., 27:275–279, 1971.
[Bir29] G. D. Birkhoff. Demonstration d’un theoreme elementaire sur les fonctions entieres. C. R. Acad. Sci. Paris, 189:473–475, 1929.
[Bos84] Michael Boshernitzan. A unique ergodicity of minimal symbolic flows with linear block growth. Journal d’Analyse Mathématique, 44(1):77–96, 1984.
[Bos92] Michael D Boshernitzan. A condition for unique ergodicity of minimal symbolic flows. Ergodic Theory and Dynamical Systems, 12(3):425–428, 1992.
[BS97] Paul S. Bourdon and Joel H. Shapiro. Cyclic phenomena for composition operators. Mem. Amer. Math. Soc., 125(596):x+105, 1997.
[BS22] David Burguet and Ruxi Shi. Topological mean dimension of induced systems. 2022.
[CK19] Van Cyr and Bryna Kra. Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc. (JEMS), 21(2):355–380, 2019.
[Coo15] Michel Coornaert. Topological dimension and dynamical systems. Springer, 2015.
[CP23] Darren Creutz and Ronnie Pavlov. Low complexity subshifts have discrete spectrum. arXiv preprint arXiv:2302.10336, 2023.
[Cre23] Darren Creutz. Measure-theoretically mixing subshifts of minimal word complexity, 2023.
[Dan13] Alexandre I. Danilenko. A survey on spectral multiplicities of ergodic actions. Ergodic Theory Dynam. Systems, 33(1):81–117, 2013.
[DDMP21] Sebastián Donoso, Fabien Durand, Alejandro Maass, and Samuel Petite. Interplay between finite topological rank minimal cantor systems, s-adic subshifts and their complexity. Transactions of the American Mathematical Society, 374(5):3453–3489, 2021.
[DGS06] Manfred Denker, Christian Grillenberger, and Karl Sigmund. Ergodic theory on compact spaces, volume 527. Springer, 2006.
[DP22] Fabien Durand and Dominique Perrin. Dimension groups and dynamical systems—substitutions, Bratteli diagrams and Cantor systems, volume 196 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2022.
[Esp23] Bastián Espinoza. The structure of low complexity subshifts, 2023.
[Fog02] N. Pytheas Fogg. Substitutions in dynamics, arithmetics and combinatorics, volume 1794 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002. Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel.
[GT20] Yonatan Gutman and Masaki Tsukamoto. Embedding minimal dynamical systems into hilbert cubes. Inventiones mathematicae, 221(1):113–166, 2020.
[GW95] Eli Glasner and Benjamin Weiss. Quasi-factors of zero-entropy systems. Journal of the American Mathematical Society, 8(3):665–686, 1995.
[Jav19] Mohammad Javaheri. Cyclic composition operators on separable locally compact metric spaces. Topology Appl., 258:126–141, 2019.
[Kat01] Anatole Katok. Cocycles, cohomology and combinatorial constructions in ergodic theory. In Smooth ergodic theory and its applications (Seattle, WA, 1999), volume 69 of Proc. Sympos. Pure Math., pages 107–173. Amer. Math. Soc., Providence, RI, 2001. In collaboration with E. A. Robinson, Jr.
[KL97] J. Kwiatkowski and Y. Lacroix. Multiplicity, rank pairs. J. Anal. Math., 71:205–235, 1997.
[Lin99] Elon Lindenstrauss. Mean dimension, small entropy factors and an embedding theorem. Inst. Hautes Études Sci. Publ. Math., 89(1):227–262, 1999.
[LT14] Elon Lindenstrauss and Masaki Tsukamoto. Mean dimension and an embedding problem: an example. Israel Journal of Mathematics, 199(2):573–584, 2014.
[LV97] Pierre Liardet and Dalibor Volnỳ. Sums of continuous and differentiable functions in dynamical systems. Israel Journal of Mathematics, 98:29–60, 1997.
[Mos96] Brigitte Mossé. Recognizability of substitutions and complexity of automatic sequences. Bull. Soc. Math. Fr., 124(2):329–346, 1996.
[NP66] D. Newton and W. Parry. On a factor automorphism of a normal dynamical system. Ann. Math. Statist., 37:1528–1533, 1966.
[Par53] O. S. Parasyuk. Flows of horocycles on surfaces of constant negative curvature. Uspehi Matem. Nauk (N.S.), 8(3(55)):125–126, 1953.
[Que10] Martine Queffélec. Substitution dynamical systems-spectral analysis, volume 1294. Springer, 2010.

		$\displaystyle\\|h-\sum_{l,n}a_{l,n}\tilde{g_{l}}\circ f^{2n}-\sum_{l,n}b_{l,n}\tilde{g_{l}}\circ f^{2n+1}\\|_{[a,1],\infty}$
	$\displaystyle=$	$\displaystyle\\|h-\sum_{l,n}a_{l,n}{g_{l}}(a)-\sum_{l,n}b_{l,n}{g_{l}}\circ f^{2n+1}\\|_{[a,1]\infty}$
	$\displaystyle\leq$	$\displaystyle\|h(a)-\sum_{l,n}a_{l,n}{g_{l}}(a)\|+\\|h-\sum_{l,n}b_{l,n}{g_{l}}\circ f^{2n+1}-h(a)\\|_{[a,1]\infty}<2\epsilon.$

(7 $\cdot$ 3)	$\displaystyle N(u\|x_{1}^{p})$	$\displaystyle\geq N(u\|w_{n})\frac{N(w_{n}\|x_{1}^{p})}{L_{n}},$
	$\displaystyle\geq N(u\|v_{n})K_{n}\frac{N(w_{n}\|x_{1}^{p})}{L_{n}},$
	$\displaystyle\geq N(u\|v_{n})N(w_{n}\|x_{1}^{p}).$

	$\displaystyle d(u\|w_{n})\geq$	$\displaystyle d(u\|v_{n})\frac{K_{n}\|v_{n}\|}{(K_{n}+1)\|v_{n}\|},$
	$\displaystyle\geq$	$\displaystyle\frac{1}{2}d(u\|v_{n}).$

Multiplicity of topological systems

Abstract.

2020 Mathematics Subject Classification:

1. Introduction

Theorem.

Theorem.

2. Topological multiplicity, definition and first properties

2.1. Multiplicity of a linear operator

Lemma 2.1.

Proof.

2.2. Operator of composition : Topological and ergodic multiplicities.

Definition 2.2.

Definition 2.3.

Lemma 2.4.

Proof.

Question 2.5.

2.3. The number of ergodic measures as a multiplicity.

Lemma 2.6.

Proof.

Remark 2.7.

Corollary 2.8.

2.4. Relating Mult​(T)\text{\rm Mult}(T) and Mult​(U¯T)\text{\rm Mult}(\underline{U}_{T})

Proposition 2.9.

Lemma 2.10.

Proof.

Proof of Proposition 2.9.

3. Affine embedding of (ℳ​(X),T∗)(\mathcal{M}(X),T_{*}) in cubical shifts

3.1. Case of finite sets

Proposition 3.1.

Proof.

Remark 3.2.

3.1.1. General case

Theorem 3.3.

Proof.

3.2. Affine embeddings and Lindenstrauss-Tsukamoto conjecture

Corollary 3.4.

Proof.

3.3. Application: zero topological entropy

Proposition 3.5.

Theorem 3.6.

Topological proof of Proposition 3.5.

4. Baxter’s Lemma in Banach spaces

Lemma 4.1.

Proof.

Claim 4.2.

Proof of Claim 4.2.

5. Cantor systems with finite topological rank

Theorem 5.1.

Proof.

Remark 5.2.

Question 5.3.

6. Examples of finite topological multiplicity

6.1. Minimal rotation on compact groups

Proposition 6.1.

Proof.

6.2. Sturmian subshift

Proposition 6.2.

Proof.

6.3. Homeomorphism of the interval

Theorem 6.3.

Lemma 6.4.

Proof.

Proposition 6.5.

Proof.

Question 6.6.

6.4. Substitution Subshift

Theorem 6.7.

Theorem 6.8.

Lemma 6.9.

Proof.

Lemma 6.10.

Proof.

Proof of Theorem 6.9.

Question 6.11.

7. Subshifts with linear growth complexity

Theorem 7.1.

Question 7.2.

Lemma 7.3.

Proof.

Lemma 7.4 ([Bos84], Lemma 4.1).

2.4. Relating $\text{\rm Mult}(T)$ and $\text{\rm Mult}(\underline{U}_{T})$

3. Affine embedding of $(\mathcal{M}(X),T_{*})$ in cubical shifts