Ergodicity via continuity

TL;DR

This paper establishes a novel characterization of ergodicity for aperiodic automorphisms of Lebesgue spaces through the continuity of a specific map on a metric Boolean algebra, also extending to periodic and totally ergodic cases.

Contribution

It introduces a new equivalence between ergodicity and map continuity on Boolean algebras, providing a fresh perspective on ergodic theory.

Findings

Ergodicity is equivalent to the continuity of a certain map on a metric Boolean algebra.

A related characterization is provided for periodic and totally ergodic transformations.

The results unify different types of transformations under a common framework.

Abstract

We show that the ergodicity of an aperiodic automorphism of a Lebesgue space is equivalent to the continuity of a certain map on a metric Boolean algebra. A related characterization is also presented for periodic and totally ergodic transformations

Equations

$$d([A],[B])=\mu (A△B).$$

$$\phi^{(m)}_{T}([A])=\mu\Bigl{(}\bigcup_{n=0}^{m}T^{n}A\Bigr{)}.$$

$$\phi_{T}([A])=\lim_{m\to\infty}\phi^{(m)}_{T}([A])=\mu\Bigl{(}\bigcup_{n\geq 0}T^{n}A\Bigr{)}.$$

$$\Bigl{(}\bigcup_{n=0}^{m}T^{n}A\Bigr{)}\triangle\Bigl{(}\bigcup_{n=0}^{m}T^{n}B\Bigr{)}\subseteq\bigcup_{n=0}^{m}(T^{n}A\triangle T^{n}B),$$

$$|{\varphi }_T^{(m)}([A])$$

$$\displaystyle\leq\mu\Bigl{(}\bigcup_{n=0}^{m}(T^{n}A\triangle T^{n}B)\Bigr{)}\leq\sum_{k=0}^{m}\mu(T^{n}(A\triangle B))=(m+1)\mu(A\triangle B).$$

$$\Omega =\underset{n\ge 0}{⨆}{P}_n.$$

$$\sum _{n=n_0+1}^{\infty }\mu (P_n)<\epsilon /2.$$

$$A_{n_{0}}=A\bigcap\Bigl{(}\bigcup_{n=1}^{n_{0}}P_{n}\Bigr{)},$$

$${\varphi }_T([A])={\varphi }_T([A_{n_0}])+{\varphi }_T([A\setminus A_{n_0}])={\varphi }_T^{(n_0!-1)}([A_{n_0}])+{\varphi }_T([A\setminus A_{n_0}]).$$

$$d([A],[B])=\mu (A_{n_0}△B_{n_0})+\mu ((A\setminus A_{n_0}))△(B\setminus B_{n_0})<\epsilon /2n_0!,$$

$$|{\varphi }_T([A])-{\varphi }_T([B])|$$

$$\displaystyle+\mu\Bigl{(}\bigcup_{n\geq 0}T^{n}(A\setminus A_{n_{0}})\triangle\bigcup_{n\geq 0}T^{n}(B\setminus B_{n_{0}})\Bigr{)}\leq$$

$$\displaystyle<\varepsilon/2+\mu\Bigl{(}\bigcup_{n=n_{0}+1}^{\infty}P_{n}\Bigr{)}<\varepsilon/2+\varepsilon/2=\varepsilon.$$

$$\Omega =\bigcup _{\alpha \in I}{\Omega }_{\alpha }$$

$$A=\bigcup _{\alpha \in I}{A}_{\alpha },A_{\alpha }=A\cap {\Omega }_{\alpha },$$

$$\mu\Bigl{(}\bigcup_{\alpha\in I}\Omega_{\alpha}\Bigr{)}=1=\phi_{T}([A])=\mu\Bigl{(}\bigcup_{n\geq 0}T^{n}A\Bigr{)}=\mu\Bigl{(}\bigcup_{n\geq 0}T^{n}\bigcup_{\alpha\in I}A_{\alpha}\Bigr{)}=\mu\Bigl{(}\bigcup_{\alpha\in I}\bigcup_{n\geq 0}T^{n}A_{\alpha}\Bigr{)}.$$

$$0=\mu\Bigl{(}\Bigl{(}\bigcup_{\alpha\in I}\Omega_{\alpha}\Bigr{)}\setminus\bigcup_{\alpha\in I}\Bigl{(}\bigcup_{n\geq 0}T^{n}A_{\alpha}\Bigr{)}\Bigr{)}=\mu\Bigl{(}\bigcup_{\alpha\in I}\Bigl{(}\Omega_{\alpha}\setminus\Bigl{(}\bigcup_{n\geq 0}T^{n}A_{\alpha}\Bigr{)}\Bigr{)}\Bigr{)}.$$

$$\mu\Bigl{(}\bigcup_{\alpha\in J}\Omega_{\alpha}\setminus\Bigl{(}\bigcup_{n\geq 0}T^{n}A_{\alpha}\Bigr{)}\Bigr{)}=0.$$

$$\mu\Bigl{(}\bigcup_{\alpha\in J}\Omega_{\alpha}\Bigr{)}=\mu\Bigl{(}\bigcup_{\alpha\in J}\bigcup_{n\geq 0}T^{n}A_{\alpha}\Bigr{)}.$$

$$\nu_{C}(J)=\mu\Bigl{(}\bigcup_{\alpha\in J}C_{\alpha}\Bigr{)},\ \ J\subseteq I.$$

$${\nu }_A(J)=0,\text{but}{\nu }_{\Omega }(J)>0.$$

$$\displaystyle 0=\mu\Bigl{(}\bigcup_{n\geq 0}T^{n}\Bigl{(}\bigcup_{\alpha\in J}A_{\alpha}\Bigr{)}\Bigr{)}=\mu\Bigl{(}\bigcup_{\alpha\in J}\Bigl{(}\bigcup_{n\geq 0}T^{n}A_{\alpha}\Bigr{)}\Bigr{)}=\mu\Bigl{(}\bigcup_{\alpha\in J}\Omega_{\alpha}\Bigr{)}=\nu_{\Omega}(J)>0,$$

$$\bigcup _{\alpha \in J}{\Omega }_{\alpha }=\bigcup _{n\in \mathbb{Z}}{T}^{n}B(\mathrm{mod}\mu ).$$

$$\mu\Bigl{(}\bigcup_{\alpha\in J}\Omega_{\alpha}\Bigr{)}=\mu\Bigl{(}\bigcup_{n\in\mathbb{Z}}T^{n}B\Bigr{)}=\mu\Bigl{(}\bigcup_{n\geq 0}T^{n}B\Bigr{)}$$

$$\mu\Bigl{(}\bigcup_{\alpha\in I\setminus J}B_{\alpha}\Bigr{)}=0.$$

$$d([A],[B])$$

$$\displaystyle=\mu\Bigl{(}\bigcup_{\alpha\in I\setminus J}A_{\alpha}\Bigr{)}+\mu\Bigl{(}\bigcup_{\alpha\in J}(A_{\alpha}\triangle B_{\alpha})\Bigr{)}=\nu_{A}(I\setminus J)+\mu\Bigl{(}\bigcup_{\alpha\in J}(A_{\alpha}\triangle B_{\alpha})\Bigr{)}<\delta,$$

$${\varphi }_T([A])-{\varphi }_T([B])$$

$$\displaystyle=\mu\Bigl{(}\bigcup_{\alpha\in I}\Omega_{\alpha}\Bigr{)}-\mu\Bigl{(}\bigcup_{\alpha\in J}\Omega_{\alpha}\Bigr{)}=$$

$$={\nu }_{\Omega }(I\setminus J)<\epsilon ,$$

$${\mu_{\Omega\setminus B}\Bigl{(}\bigcup_{k=0}^{n_{0}-1}T^{k}E\Bigr{)}>1-\varepsilon}.$$

$$d([A],[C])=\mu (A△C)=\mu (E)<\frac{1}{n_0}\mu (\Omega \setminus B)$$

Full text

MSC2010: 37A25; 28D05; 54C05

1 Introduction

The main goal of this short note is to show that the ergodicity of an aperiodic transformation $T$ of a Lebesgue space ${(\Omega,\mathfrak{F},\mu)}$ is equivalent to the continuity of a certain transformation associated with $T$. There are many different but equivalent definitions of ergodicity for measure preserving transformations in the literature (see [2, §2.3] for example). Yet another criterion presented here seems to be new and quite interesting.

Let ${(\mathcal{F},d)}$ be the metric space of $\mu$-equivalent classes of $\mathfrak{F}$-measurable sets (a set ${A\in\mathfrak{F}}$ belongs to the class $[B]$ induced by a set ${B\in\mathfrak{F}}$ iff ${\mu(A\triangle B)=0}$). The metric $d$ is the Frechet–Nikodym metric defined as

[TABLE]

Let ${\mathcal{N}\in\mathcal{F}}$ denote the class of sets of $\mu$-measure zero. Given an automorphism $T,$ for each $m\in\mathbb{N}$ define the map ${\phi^{(m)}_{T}:\mathcal{F}\to[0,1]}$ as

[TABLE]

We also put

[TABLE]

The sequence ${\{\phi^{(m)}_{T}([A])\}_{m\in\mathbb{N}}}$ is called the wandering rate of $A$ [1, §3.8].

It is well known that $T$ is ergodic iff ${\phi_{T}([A])=1}$ for every ${A\not\in\mathcal{N}}.$ It turns out equivalent to $\phi_{T}$ being continuous everywhere except the point $\mathcal{N},$ which is the main statement (Theorem 2) in this note. We also present related characterizations for periodic (Theorem 1) and for totally ergodic transformations (Theorem 3).

2 Continuity of the maps

2.1. Periodic transformation. The first local aim is to investigate the continuity of the transformations ${\phi^{(m)}_{T}}$, for $m\in\mathbb{N}.$ It is required for studying the continuity of ${\phi_{T}}$ for periodic transformation $T.$ The continuity of ${\phi^{(m)}_{T}}$ is quite easy to prove by using the methods of university measure theory courses. We give a proof of this assertion for completeness of exposition.

Lemma 1

The transformation ${\phi^{(m)}_{T}:\mathcal{F}\to[0,1]}$ is everywhere continuous for each ${m\in\mathbb{N}}.$

**Proof **

It is evident that

[TABLE]

and then

[TABLE]

This completes the proof. It is worth noting that even the Lipschitz property of $\phi^{(m)}_{T}$ follows from the proof.

Recall the definitions of periodic and aperiodic transformations. A point ${x\in\Omega}$ is called periodic for $T$ if there exists a number ${n\in\mathbb{N}}$ with ${T^{n}x=x},$ and the smallest of these numbers is called the period of $x.$ Denote the set of periodic points of period $n\in\mathbb{N}$ by $P_{n}$ and the set of aperiodic points by $P_{0}$. It is clear that

[TABLE]

If ${\mu(P_{0})=0}$ then the automorphism $T$ is called almost everywhere periodic (or shortly periodic). If ${\mu(P_{0})=1}$ then $T$ is an aperiodic transformation.

The following proposition on the continuity of $\phi_{T}$ for periodic automorphisms $T$ is a corollary of Lemma 1.

Proposition 1

Let $T$ be a periodic automorphism of a probability space ${(\Omega,\mathfrak{F},\mu)}.$ Then the transformation $\phi_{T}$ is everywhere continuous.

**Proof **

Since ${\sum_{n=1}^{\infty}\mu(P_{n})=1},$ for arbitrary ${\varepsilon>0}$ there exists a number ${n_{0}=n_{0}(\varepsilon)\geq 1}$ such that

[TABLE]

For every ${A\in\mathfrak{F}},$ put

[TABLE]

and then

[TABLE]

As soon as

[TABLE]

the last calculation in the proof of Lemma 1 yields

[TABLE]

This completes the proof.

The converse to Proposition 1 is discussed in the next subsection.

2.2. The points of continuity. The following statements describe in detail all points of continuity of $\phi_{T}.$

Lemma 2

Let $T$ be an automorphism of a Lebesgue space ${(\Omega,\mathfrak{F},\mu)}.$ If ${\phi_{T}([A])=1}$ then $[A]$ is a continuity point of $\phi_{T}.$

Proposition 2

Suppose that $T$ is an automorphism of a Lebesgue space ${(\Omega,\mathfrak{F},\mu)}$ and ${\mu(P_{0})>0}.$ Then $\mathcal{N}$ is a discontinuity point of $\phi_{T}.$ Moreover, if $T$ is aperiodic then ${\phi_{T}([A])<1}$ iff $[A]$ is a discontinuity point of $\phi_{T}.$

Before proving these assertions, we remark that Propositions 1 and 2 together imply the following characterization of periodic automorphisms of a Lebesgue space.

Theorem 1

An automorphism $T$ of a Lebesgue space ${(\Omega,\mathfrak{F},\mu)}$ is periodic iff $\phi_{T}$ is everywhere continuous.

**Proof **** **(of Lemma 2)

Consider the partition

[TABLE]

of $\Omega$ into ergodic components ${\Omega_{\alpha}}$ where $I$ is the set of indices (see [5] for example). Express the condition  ${\phi_{T}([A])=1}$ in terms of this ergodic decomposition:

[TABLE]

and

[TABLE]

It follows that

[TABLE]

Consequently, for each ${J\subseteq I},$ we have

[TABLE]

It is equivalent to

[TABLE]

On the set $I$ of indices consider the family of measures ${\{\nu_{C}\}}_{C\in\mathfrak{F}}$ defined as

[TABLE]

We claim that (1) guarantees the equivalence of the probability measure ${\nu_{\Omega}}$ and the measure ${\nu_{A}}.$ It is clear that ${\nu_{A}\ll\nu_{\Omega}}.$ Suppose that the opposite is false. Then there exists a set ${J\subset I}$ with

[TABLE]

It follows that

[TABLE]

which is a contradiction. Consequently, for each ${\varepsilon>0}$ there exists ${\delta=\delta(A,\varepsilon)>0}$ such that ${\nu_{A}(J)<\delta}$ implies ${\nu_{\Omega}(J)<\varepsilon}.$

Now we are ready to prove the continuity of $\phi_{T}$ at the point $[A].$ For arbitrary $\varepsilon>0,$ take $\delta>0$ as in the previous discussion. For ${B\in\mathfrak{F}}$ with ${\mu(B)>0},$ because the set ${\bigcup_{n\in\mathbb{Z}}T^{n}B}$ is invariant under $T,$ there exists a set ${J=J(B)\subset I}$ of indices such that

[TABLE]

This yields

[TABLE]

and

[TABLE]

Assume that ${d([A],[B])<\delta}.$ We have

[TABLE]

which implies ${\nu_{A}(I\setminus J)<\delta}$ and, hence, ${\nu_{\Omega}(I\setminus J)<\varepsilon}.$ It follows that

[TABLE]

completing the proof.

**Proof **** **(of Proposition 2)

Without loss of generality, assume that the automorphism $T$ is aperiodic. Suppose that ${\phi_{T}([A])<1},$ and then the set ${B=\bigcup\limits_{n\in\mathbb{Z}}T^{n}A}$ of measure ${\mu(B)<1}$ is $T$-invariant.

The restriction of $T$ to ${\Omega\setminus B}$ is aperiodic and preserves the probability measure $\mu_{\Omega\setminus B.}$ Applying the Rokhlin–Halmos lemma (see [3] for example), we find that for ${\varepsilon>0}$ and $n_{0}\geq 1$ there exists a set ${E\subset\Omega\setminus B}$ such that the sets ${T^{k}E}$ for ${0\leq k\leq n_{0}-1}$ are disjoint and satisfy the inequality

[TABLE]

It is clear that $\mu_{\Omega\setminus B}(E)<\frac{1}{n_{0}}.$ Put ${C=A\cup E}.$ Then

[TABLE]

and

[TABLE]

In this way, taking ${\varepsilon=1/2}$ and sufficiently large ${n_{0}\geq 1}$ we obtain $d([A],[C])$ is small enough but

[TABLE]

This proves that $\phi_{T}$ is discontinuous at the point ${[A]}$ with ${\phi_{T}([A])<1}.$

Now, if $[A]$ is a discontinuity point of ${\phi_{T}([A])}$ then Lemma 2 tells us that ${\phi_{T}([A])<1}.$ The proof is complete.

2.3. Ergodic and totally ergodic transformations. The following characterization of ergodic transformations is also direct corollary of Proposition 2.

Theorem 2

An aperiodic automorphism $T$ of a Lebesgue space ${(\Omega,\mathfrak{F},\mu)}$ is ergodic iff ${\phi_{T}:\mathcal{F}\to[0,1]}$ is continuous everywhere except the point $\mathcal{N}$.

**Proof **

If $T$ is ergodic then ${\phi_{T}([A])=1}$ for ${\mu(A)>0}.$ Consequently, Lemma 2 shows that $\phi_{T}$ is continuous at such points. However, Proposition 2 states that $\mathcal{N}$ is a discontinuity point.

Now, if $T$ is not ergodic then there exists an $\mathfrak{F}$-measurable $T$-invariant set $A$ with ${0<\mu(A)<1}$. It follows that ${\phi_{T}([A])=\mu(A)<1}$ and Proposition 2 implies that $\phi_{T}$ has a discontinuity at $[A].$

As another application of Proposition 2, we discuss here a characterization of totally ergodic transformations, which means that the powers $T^{n}$ for all $n\in\mathbb{N}$ are ergodic transformations.

Define a new map ${\phi_{T}^{*}:\mathcal{F}\to[0,1]}$ as

[TABLE]

Theorem 3

An aperiodic automorphism $T$ of a Lebesgue space ${(\Omega,\mathfrak{F},\mu)}$ is totally ergodic iff ${\phi^{*}_{T}:\mathcal{F}\to[0,1]}$ is continuous everywhere except the point $\mathcal{N}$.

**Proof **

If $T$ is totally ergodic then $\phi_{T^{k}}([A])=1$ for all $k\geq 1$ and all ${A\not\in\mathcal{N}}.$ Hence, ${\phi^{*}([A])=1},$ and therefore $\phi^{*}$ is continuous at that point. Indeed, for arbitrary ${\varepsilon>0}$ we take ${0<\delta<\mu(A)}.$ Then the inequality ${d([A],[B])<\delta}$ implies that ${\mu(B)>0}$ and therefore

[TABLE]

It is evident, that the class $\mathcal{N}$ is a discontinuity point of ${\phi^{*}_{T}}.$

Assume now that $T$ is not totally ergodic. Take the smallest ${k_{0}\geq 1}$ such that $T^{k_{0}}$ is not ergodic. Therefore all powers $T^{nk_{0}}$ for ${n\geq 1}$ are not ergodic either (because the invariant sets of $T^{k_{0}}$ are invariant under $T^{nk_{0}}$ for all ${n\geq 1}$). It is clear that there are only finitely many, at most $k_{0},$ such sequences of non ergodic transformations ${\{T^{nk}\}_{n\geq 1}}.$ Denote by $\mathcal{K}$ the finite set of possible values of $k.$ Thus, ${k_{0}\in\mathcal{K}}$ and ${|\mathcal{K}|\leq k_{0}}.$ Put ${\kappa=\prod_{k\in\mathcal{K}}k}.$ Take a nontrivial invariant set ${A\not\in\mathcal{N}}$ of the transformation ${T^{\kappa}}.$ We claim that it is a discontinuity point of $\phi^{*}_{T}.$

By Theorem 2, all transformations $\phi_{T^{n\kappa}}$ for ${n\geq 1}$ (being non ergodic) are discontinuous at $[A].$ Considering (2), we conclude that for sufficiently small $\delta>0$ and all ${n\geq 1}$ there exist some sets ${E_{n}\subset\Omega\setminus A}$ such that ${B_{n}=A\cup E_{n}}$ satisfy

[TABLE]

For the set ${B=A\bigcup(\bigcup_{n\geq 1}E_{n})}$ it is easy to see that ${d([A],[B])=\mu(\bigcup_{n\geq 1}E_{n})<\delta},$ and for all ${n\geq 1},$

[TABLE]

Considering the value ${\phi^{*}_{T}([B])},$ we conclude that

[TABLE]

for some ${k^{\prime}=k^{\prime}(B)\in\mathcal{K}}.$ The second equality is true because for the ergodic transformation $T^{m}$ we have ${\phi_{T^{m}}([B])=1},$ and therefore the infimum is reached on non ergodic transformations.

For arbitrary $\varepsilon>0$ there exists a number ${n^{\prime}\in\mathbb{N}}$ such that

[TABLE]

Taking into account the estimates (3) and (4), the monotonicity property

[TABLE]

and the equality

[TABLE]

we obtain

[TABLE]

Take sufficiently small $\varepsilon>0$ so that the expression ${\frac{1}{2}\mu(\Omega\setminus A)-\varepsilon}$ is positive. Then the last estimate guarantees that $\phi^{*}_{T}$ is discontinuous at $[A].$ The proof is complete.

In conclusion, we remark that it would be interesting to find a related characterization for transformations with a different type of mixing property (see [4] for example).

ACKNOWLEDGMENTS. The work was supported by the program of fundamental scientific research of SB RAS № I.1.2., project № 0314-2016-0005.