Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

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2
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2answers
219 views

What is the adic realization of a Bernoulli shift ?

Roughly speaking, a theorem by Vershik says that every ergodic invertible measure-preserving transformation is isomorphic to some "adic" transformation on the spaces of paths of a Bratelli-Vershik ...
1
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0answers
172 views

Ergodicity of non-homogeneous “rotations”

It is well known that a rotation $f(z)=e^{i\theta}z$ of the unit circle, is ergodic if and only if $e^{i\theta}$ is not a root of unity. Now, what happens if we let $\theta$ depend on $z$ (say, ...
10
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0answers
484 views

Centralizers of group actions

Let a locally compact group $G$ act on a probability space $(X,\mu)$. Define the centralizer by $C(G)=\{\Delta\in Aut(X,\mu)\mid \Delta(gx)=g\Delta(x)\text{ almost everywhere}\}$. $Aut(X,\mu)$ denotes ...
4
votes
2answers
366 views

partition into the orbits of a dynamical system

Let $T$ be a measure-preserving invertible transformation of a Lebesgue space, and let $P$ be the partition of the Lebesgue space into the orbits of $T$. 1) Is it true that $P$ is nonmeasurable (in ...
5
votes
1answer
258 views

Limits of intrinsically ergodic systems

Let $(X_i)$ be a sequence of compact metric spaces and $(f_i)$ a sequence of transitive transformations $f_i:X_i \to X_i$ with $0 < h_{top}(f_i) < \infty$. The sequence of dynamical systems ...
3
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1answer
278 views

Recurrence for sets of finite measure on infinite measure space

Thanks to your helpful feedback, I have made my claim more precise. Claim Given an infinite measure space $\left( X,\mathcal B, \mu\right)$ and an ergodic, invertible, measure preserving and ...
14
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5answers
2k views

Category Theory and Ergodic Theory

I am very much interested in finding out about any category theoretical work on dynamical systems and on ergodic theory. On the face of it, it seems that a categorical language can go a long way, at ...
5
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0answers
428 views

Density of strictly ergodic measures in the d-bar topology

I am currently studying a problem which deals with cocycles of highly noncompact operators on Hilbert space, with the base transformation being the full shift on two symbols. In my particular ...
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0answers
233 views

weak star density of atomic invariant measures ?

Let us consider expanding maps $E_m: x\mapsto mx$ on the circle written additively. If we consider the set of all $E_2$ invariant probability Borel measures then the convex hull of atomic measures ...
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1answer
501 views

Shift invariant measures that are(n't) convex combinations of ergodic measures

Let $X=2^\omega = \lbrace 0,1 \rbrace^{\mathbb{N}}$ and $T\colon X \to X$ the (left) shift map. The space $\mathcal{M}$ of $T$-invariant Borel probability measures is convex with $\mathcal{M}^e$, the ...
10
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2answers
756 views

Connectedness of space of ergodic measures

Let $X = \Sigma_p^+ = \{1,\dots,p\}^\mathbb{N}$ and let $f=\sigma\colon X\to X$ be the shift map. Let $\mathcal{M}$ be the space of Borel $f$-invariant probability measures on $X$ endowed with the ...
2
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1answer
277 views

Measure of large cylinder sets

Given an ergodic measure m on a shift space, by Shannon-Mcmillan-Breiman Theorem, up to at most an $\epsilon$-portion, all cylinder sets of length $n$ (large enough) have $m$-measure between ...
3
votes
2answers
264 views

Cesaro bounded Operator which is not power bounded

Good evening! Let X be a banachspace and T a bounded linear operator on X. The cesaro avearges of T are defined as: $A_n:=\frac{1}{n} \sum\limits_{j=0}^{n-1}T^j $ We call T cesaro bounded if: ...
3
votes
1answer
224 views

The average recurrence time

As seen on wikipedia, given a measure space $(X,\Sigma,\mu)$ with $\mu(X) < \infty$ and a measure preserving transformation $T: X \mapsto X$. Let $A \subset X$ be a set of positive measure. Define ...
4
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1answer
302 views

locally-free Lie group action not preserving any measure

I'd like to know if there exists a connected Lie group $G$ and a closed manifold $M$ such that there is a locally-free smooth action $G\times M\to M$ (i.e. the stabilizer of any point of $M$ is a ...
12
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9answers
3k views

Book recommendation for ergodic theory and/or topological dynamics?

Hello, I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well ...
2
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0answers
409 views

Virtual nilpotent ergodic average

In a recent paper of Miguel N. Walsh,"Norm convergence of nilpotent ergodic averages"(http://arxiv.org/abs/1109.2922v2),the author gives a proof of the fact that multiple polynomial ergodic averages ...
3
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2answers
1k views

Shannon-McMillan-Breiman Theorem

Does anyone know of an easy proof of Shannon-McMillan-Brieman Theorem? Thanks
8
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3answers
488 views

non-integrable subadditive ergodic theorem

Dear MO_World, I have (another) question about relaxing the assumptions in the sub-additive ergodic theorem. Apologies if this is something I should know already... There are a number of statements ...
8
votes
2answers
740 views

Fourier transform of x2 invariant measure

Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...
1
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0answers
292 views

What is the mean-value of a particular exponential sum related to the non-trivial zeros of Riemann's zeta function?

This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...
14
votes
1answer
924 views

Rokhlin lemma for arbitrary infinite groups.

Let $G$ be an at most countable discrete group acting freely on a standard probability measure space $X$ in a measure preserving way. It is well known that if $G$ is a finite group then this action ...
3
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1answer
242 views

Topological weak mixing and $\omega$-linearly-independent sequences generated by composition operators

A research problem on which I am currently working requires a construction in topological dynamics of the following type: Let $T \colon X \to X$ be a continuous transformation of a compact metric ...
3
votes
2answers
338 views

Margulis-Ruelle inequality for piecewise continuous interval maps

The Margulis-Ruelle inequality states that measure-theoretic entropy is controlled by Lyapunov exponents; more precisely, if $f$ is a $C^{1+\alpha}$ diffeomorphism on a $d$-dimensional manifold $M$ ...
9
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2answers
539 views

Name this periodic tiling

Hello MO, I've been working on a problem I'm working on in ergodic theory (finding Alpern lemmas for measure-preserving $\mathbb R^d$ actions) and have found some neat tilings, that I presume were ...
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0answers
95 views

Finitary factors of Bernoulli schemes that pair duals

This question is related to my question: entropy preserving finitary factor maps of Bernoulli schemes. Hopefully, this one is a bit easier. Let $X=\{0,1\}^\mathbb{Z}$ with measure ...
2
votes
1answer
232 views

Compact group extension of a zero entropy system.

Suppose $T: X \to X$ is a continuous map and $\mu$ a $T$-ergodic probability measure over the Borel sets of $X$. Now, suppose $K \subset \mathrm{Hom}(X)$ is a compact group of measure-preserving ...
10
votes
1answer
1k views

System with invariant measure, but no ergodic measure.

Question Examples of continuous transformations $T: X \to X$ such that the family of invariant probability measures $M(T)$ is NOT empty but there is no ergodic measure ($E(T) = \emptyset$). Notice ...
3
votes
1answer
431 views

Furstenberg-Zimmer Theorem: non-invertible systems.

Questions Is there a version of Furstenber-Zimmer Theorem for non-invertible measure preserving systems? Where can I find it? What is the precise statement? Background In many works that ...
10
votes
3answers
744 views

Supremum amongst Kolmogorov-Sinai entropies: ergodic or just invariant measures.

Cases where $sup_{\mu \in E(T)} h_\mu(T) \neq \sup_{\mu \in M(T)} h_\mu(T)$. Background For a topological space $X$, let $T: X \to X$ be a continuous application. Then, call the set of ...
0
votes
1answer
163 views

Refining ladders and orbit segments - with a picture

I am trying to understand the following paragraph from The Classification of Non-Singular Actions, Revisited, page 5 paragraph 2. Remember that $S \in [T]$ so that for every $x\in X, S(x) = ...
3
votes
2answers
336 views

How to detect frequency?

Let $J$ be an arc in $\mathbb{S}^{1}\subset\mathbb{C}$ (no matter open or closed) and $\alpha\in(0,2\pi)$ be an angle such that $\alpha/\pi$ is irrational. Consider in $\mathbb{S}^{1}$ the sequence ...
4
votes
1answer
163 views

Subadditive Kingmans theorem for lattices.

I am looking for a multidimensional version of Kingman's subadditive theorem. I found this but it is not exactely what I need. I would rather have something like that: Let us consider ...
10
votes
1answer
697 views

Different uses of the word “ergodic”

There appear to be two definitions of the word ergodic. The dynamical systems definition says that a measure space $(X,\mathit B, \mu)$ and measure preserving transformation $T: X \mapsto X$ is ...
3
votes
1answer
420 views

Example of a measure-preserving system with dense orbits that is not ergodic

Let $X$ be a Borel probability space (i.e. equipped with a measure $\mu$ on the Borel $\sigma$-algebra such that $\mu(X) = 1$) with a measure-preserving transformation $T$ such that every point has a ...
3
votes
1answer
212 views

Accumulation points of the Birkhoff average of $m$

Let $M$ be a closed manifold, $m$ be the normalized volume measure on $M$, and $f:M\to M$ be a $C^2$ transitive Anosov diffeomorphism. Consider the pushforward $f^km$ defined by ...
14
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3answers
894 views

Applications of and motivation for von Neumann's mean ergodic theorem

I stated von Neumann's mean ergodic theorem (VNMET) in a talk recently and someone in the audience asked what it was good for. The only application I knew of VNMET was to prove Birkhoff's ergodic ...
2
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3answers
255 views

How long does it take a Brownian particle to achieve a uniform probability distribution across a space?

Imagine I have a point-like Brownian particle, with diffusion constant $D$, and I place it at some initial coordinate in a cage of known geometry. Assuming the volume $V$ of the cage is "everywhere" ...
11
votes
3answers
874 views

Alternative proofs of the Krylov-Bogolioubov theorem

The Krylov-Bogolioubov theorem is a fundamental result in the ergodic theory of dynamical systems which is typically stated as follows: if $T$ is a continuous transformation of a nonempty compact ...
6
votes
4answers
546 views

A follow up question related to entropy

For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each ...
11
votes
1answer
420 views

Measure conjugacy and ergodic decomposition

Roughly speaking, this question asks whether there is a measure-conjugacy between two transformations if there are measure-conjugacies between their ergodic components. Suppose $(X,\mu)$ is a ...
7
votes
1answer
229 views

Non-oscillatory behaviour in the subadditive ergodic theorem

I am currently reading an article in which the author goes to certain lengths which could be avoided if the following result were true: Lemma (proposed): Let $T$ be an ergodic ...
1
vote
1answer
269 views

Does specification implies that entropy map is upper semicontinuous?

Let $(X,d)$ be a compact metric space and f a continuous transformation on X. f has the specification if one can always find a single orbit to interpolate between different pieces of orbits, up to a ...
3
votes
0answers
250 views

A general Lipschtiz potential can be specified by a Gibbs specification ?

I want to consider one-dimensional system on the lattice $\mathbb{L}=\mathbb{N}$. Let be $A:(\mathbb{S}^1)^{\mathbb{L}}\to\mathbb{R}$ a lipschtiz potential. Consider the Ruelle operator $$ ...
12
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5answers
2k views

Proof of Krylov-Bogoliubov Theorem

Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
31
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2answers
2k views

Ergodic Theorem and Nonstandard Analysis

Here is a quote from Lectures on Ergodic Theory by Halmos: I cannot resist the temptation of concluding these comments with an alternative "proof" of the ergodic theorem. If $f$ is a complex ...
21
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2answers
842 views

Billiard dynamics for multiple balls

I am interested to learn to what extent results on billiards in polygons have been extended to multiple balls. Assume the balls have equal radii and the same mass, the same initial speed, and all ...
6
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0answers
302 views

“topological” conjugacy of group automorphisms

In the paper "Orbit Equivalence and Topological Conjugacy of Affine Actions on Compact Abelian Groups", S. Bhattacharya shows (Theorem 3) the following: Theorem. Given two actions $\alpha$ and ...
3
votes
1answer
437 views

Ergodicity of Convoluted White Noise

I have a question regarding ergodicity in infinite dimensional spaces. Let $\mathcal{D}$ be the space of distributions on a Schwartz space, and let $\mu$ be the white noise process which exists by ...
2
votes
1answer
208 views

Multiple ergodic averages with varying number of terms

Hi. I've been stuck on the following question for some time. Consider a sequence of functions $\left( f_n \right)$ from an ergodic space $\left( \mathsf{X}, \mathsf{S}, \mu \right)$ to $\left[ 0,1 ...