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

**2**

votes

**0**answers

189 views

### The ergodic theorem and Lorentz norms

Let $\Omega$ be a probability space, and $\{ \tau \}_{y\in \mathbb R^d}$ an ergodic group of measure-preserving transformations, $\tau_y:\Omega \to \Omega$.
The ergodic theorem says that if $f \in ...

**5**

votes

**0**answers

205 views

### Can invariant means be really considered as the generalization of the uniform measure?

I am writing a paper for game theorists where I use (countable) amenable groups to do some things. So I am writing up a preliminary section about countable amenable groups whose main purpose is to ...

**3**

votes

**1**answer

246 views

### Understanding a proof that the simplex of shift invariant probability measures on $\{0,1\}^\mathbb{Z}$ is Poulsen?

This is a question on the proof of this fact in chapter 3 of the book "Functional Analysis: Surveys and Recent Results II". There at the end the proof is outlined as follows:
Let $\mu$ be a shift ...

**7**

votes

**0**answers

226 views

### resampling over Bowen balls

Hello MO World
I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and ...

**1**

vote

**0**answers

160 views

### Entropy of Bernoulli walks on semi-groups.

Consider the Fibonacci semi-group $<L,R|LRR=RLL>$ with a Bernoulli walk $P(R)=p, P(L)=1-p$. Is the entropy $H(p)$ an unimodal function with maximum at p=0.5? Is this true for all finitely ...

**1**

vote

**1**answer

144 views

### ergodicity of the group of transformations preserving a partition

Let $X=\{0,1\}^{\mathbb{N}}$ and $\theta$ be the partition of $X$ induced by the equivalence relation $x \sim x'$ when $x$ and $x'$ differ only at a finite number of coordinates (see this related ...

**0**

votes

**1**answer

117 views

### intersection partition as an orbital partition

Let $X=\{0,1\}^{\mathbb{N}}$ and $\xi_n$ be the partition of $X$ defined by the equivalence relation $x \sim_n x' \Leftrightarrow (x_{n}, x_{n+1}, \ldots) = (x_{n}', x_{n+1}', \ldots)$. The sequence ...

**2**

votes

**1**answer

285 views

### functions whose average along orbits is zero or a constant

Is there some name in ergodic theory or integrable systems theory for a function whose average value on every orbit is zero? (Of course when I say "every orbit" in the context of ergodic theory I ...

**7**

votes

**2**answers

448 views

### Generalization(s) of Subadditive Ergodic Theorems

I am interested in dynamical gadgets which can be described by sampling along the orbits of points in some ergodic system $(\Omega,\mu,T)$. When $\mu$ is a probability measure, the theory of such ...

**12**

votes

**3**answers

661 views

### Looking for at least one beautiful and not too technical result in asymptotic group theory

We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this ...

**6**

votes

**2**answers

754 views

### De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales

De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\infty$ that is ...

**4**

votes

**2**answers

483 views

### “Uncertainty principle” for self-adjoint operators in a finite von Neumann algebra

Let $M\subset B(\mathcal H)$ be a finite von Neumann algebra of bounded operators on a Hilbert space $\mathcal H$., let $P\in M$ be a self-adjoint operator with a pure-point spectrum (for example a ...

**1**

vote

**0**answers

250 views

### weak mixing and spectral theorem

I'm trying to prove an equivalent statement about weak mixing transformations that relies on the spectral theorem, but I can't find a reference to fill in the last details. A hint for solving it or ...

**3**

votes

**1**answer

311 views

### algebraic VS topological ergodicity

Let A be a $C^*$-algebra with unit $I$, and G a locally compact (Hausdorff) group. An action $\alpha$ of G on A is a strongly continuous homomorphism of G into Aut(A), the group of *-automorphisms of ...

**2**

votes

**0**answers

79 views

### adic periodic approximation of a dynamical system

Let $P$ be the Pascal adic transformation. The cutting and stacking construction of $P$ corresponds to a ``Pascal periodic approximation'' of $P$: a sequence $(P_n)$ of periodic automorphisms strongly ...

**2**

votes

**2**answers

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**

vote

**0**answers

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**

votes

**0**answers

492 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

**2**answers

375 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

**1**answer

262 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**

votes

**1**answer

281 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**

votes

**5**answers

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**

votes

**0**answers

439 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 ...

**1**

vote

**0**answers

243 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 ...

**1**

vote

**1**answer

530 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**

votes

**2**answers

789 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**

votes

**1**answer

285 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

**2**answers

270 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

**1**answer

225 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**

votes

**1**answer

304 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**

votes

**9**answers

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**

votes

**0**answers

415 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

**2**answers

1k views

### Shannon-McMillan-Breiman Theorem

Does anyone know of an easy proof of Shannon-McMillan-Brieman Theorem?
Thanks

**8**

votes

**3**answers

495 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

**2**answers

766 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

**0**answers

296 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

**1**answer

976 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

**1**answer

245 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

**2**answers

359 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

**2**answers

543 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

**0**answers

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

**1**answer

234 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

**1**answer

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

**1**answer

442 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

**3**answers

765 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

**1**answer

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

**2**answers

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

**1**answer

164 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

**1**answer

728 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

**1**answer

435 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 ...