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

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11
votes
9answers
2k 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
votes
0answers
391 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
votes
2answers
892 views

Shannon-McMillan-Breiman Theorem

Does anyone know of an easy proof of Shannon-McMillan-Brieman Theorem? Thanks
8
votes
3answers
462 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
678 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
vote
0answers
280 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 ...
13
votes
1answer
806 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
votes
1answer
226 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
312 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
votes
2answers
525 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 ...
1
vote
0answers
93 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
207 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
411 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
709 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
161 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
334 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
162 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 ...
9
votes
1answer
612 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
361 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
198 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 ...
12
votes
3answers
796 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
votes
3answers
241 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" ...
10
votes
3answers
793 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
524 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
401 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 ...
6
votes
1answer
221 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
0answers
186 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
249 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
votes
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
votes
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 ...
20
votes
2answers
785 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
votes
0answers
290 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
415 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
206 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 ...
7
votes
2answers
501 views

Is this ergodic inequality true?

Is anything similar to the following inequality true, $\displaystyle P\{\max_{n \leq k \leq m} |A_k f - A_n f| > \epsilon\} \leq C \frac{||A_m f - A_n f||_1}{\epsilon}$ where $A_n f = ...
10
votes
2answers
721 views

Non-integrable ergodic theory

Can anyone help me out with proofs/counterexamples? I'm working on an operator-valued multiplicative ergodic theorem and need what may(?) be a well-known fact. This fact (if true) would help me get ...
3
votes
1answer
290 views

trivial map on $\sigma-$algebra $\mod{}0$ is trivial

Hi everyone! I am currently studying the basic theory of measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not ...
6
votes
1answer
662 views

Entropy of first return map and suspension flows

There are some well know formulas of Abramov about derived systems. Firstly let $(X,\mu,f)$ be a probability preserving system and $A\subset X$ is measurable such that $\bigcup_{n\ge0}f^nA=X$. Let ...
3
votes
1answer
308 views

Picking a representative in a continuous way

I'm hoping for some ideas/pointers here. I'm experimenting with a Livschitz theorem for functions on a locally compact Abelian group, where the periodic orbit sums take values in a closed subgroup. ...
2
votes
1answer
659 views

Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant?

It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's ...
6
votes
3answers
990 views

amenable equivalence relation generated by an action of a non-amenable group

Question. Give a (possibly elementary) example of a probability measure preserving action $\rho\colon G \curvearrowright X$ of a finitely-generated discrete group $G$ on a standard borel space $X$ ...
3
votes
1answer
519 views

Example of a non-normal infinite index subgroup of a non-amenable group with certain properties.

This is an improved version of my previous question, where I forgot to put one of the assumptions. Question. Let $G$ be a finitely generated non-amenable discrete group, and $H$ be a subgroup of ...
14
votes
11answers
3k views

Importance of Poincaré recurrence theorem? Any example?

Recently I am learning ergodic theory and reading several books about it. Usually Poincaré recurrence theorem is stated and proved before ergodicity and ergodic theorems. But ergodic theorem does not ...
6
votes
2answers
673 views

[automatic continuity] measurable homomorphisms of (C,+)-->(C,+) or (C,+)-->(C,*) are continuous and admit an explicit description ?

I am interested in generalisation of the following fact [known as automatic continuity, as I have been pointed out below]. I am especially looking for references to papers dating back to 1920s---I ...
5
votes
2answers
264 views

Entropy of nested compact invariant sets

Let $f$ be a homeomorphism on a compact metric space $X$. $K_1\supset K_2\supset\cdots \supset K$ are compact subsets of $X$ such that $f(K_n)=K_n$ and $K=\bigcap K_n$. If $h(f, K_1)<\infty$, do we ...
17
votes
1answer
942 views

Generic points and local entropies

Let $X=\{1,\dots,p\}^\mathbb{N}$ be the space of sequences on a finite alphabet with a metric inducing the product topology, and let $\sigma\colon X\to X$ be the shift map. Let $\mu$ be a ...
43
votes
0answers
2k views

Alternating colors on a line: infinitely often or converge?

Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / ...). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...
4
votes
0answers
368 views

entropy preserving finitary factor maps of Bernoulli schemes

Let $X=\{0,1\}^\mathbb{Z}$ with measure $\mu=(p,1-p)^{\mathbb{Z}}$. Let $(\phi(x))_i=(x_i+x_{i+1})$mod$2$. If $p=1/2$, then $\phi(X)=X$. If $p \not = 1/2$, then $\phi(X)$ is not a Bernoulli scheme ...
4
votes
1answer
307 views

Question about an early result on the mixing of geodesic flows

Let $T_t$ be the geodesic flow on a surface $S$ of constant negative curvature, and let $M(f,t) := \langle \bar f \cdot (f \circ T_t) \rangle$, where $\langle f \rangle := \int_S f(x) d\mu(x)$ and ...