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

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468 views

### Connection between properties of Dynamical and Ergodic Systems

Hi All
While studying Topological and Ergodic Dynamics, I've got quite preplexed by the different Properties a system might have (minimality, regionally recurring, transitivity, mixing, ergodic, ...

**2**

votes

**1**answer

180 views

### Absolutely Continuous Invariant Measures for Piecewise Convex Maps

Hi all,
I'm interested in a class of 'generalised tent maps' $f:[0,1]\to[0,1]$ for which
1) $f$ is strictly increasing on $[0, \frac{1}{2}]$, $f(0)=0$ and $f(\frac{1}{2})=1$
2) $f$ is symmetric ...

**1**

vote

**1**answer

254 views

### Weak convergence, and Cesaro convergence (of mu_n (E) ) imply convergence (of mu_n (E))?

Let $X$ be a compact metric Borel space. Suppose $\mu_{n}(A)\rightarrow\mu(A)$
for all $\mu-$continuity sets $A$ (sets with zero boundary measure), where $\mu_{n}$ is a sequence of probability ...

**5**

votes

**1**answer

309 views

### Pesin Entropy Formula

In the form that I've seen it stated, the Pesin entropy formula states that if $M$ is a compact Riemannian manifold and $f$ is a $C^{1+\alpha}$ diffeomorphism of $M$ that preserves smooth invariant ...

**2**

votes

**1**answer

246 views

### Continuity of relative entropy with respect to the weak* topology

Let $X$ be a measurable space, and let $T$ be a measurable transformation $T:X \to X$. Let $\mathcal{P}(X)$ be the space of probability measures on $X$, equipped with the weak* topology. Define the ...

**1**

vote

**1**answer

334 views

### is the limit of ergodic functions still ergodic?

under what conditions is the limit of a sequence of ergodic functions still ergodic? are there simple counter-examples to this general statement?

**8**

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**2**answers

291 views

### What time does it take for irrational rotations to hit an interval?

Hi,
Consider $\theta_n = (\theta_0 + n \theta) \mod 1$, $\theta$ being an irrational number, and $\theta_0$ an uniform random variable in $(0,1)$. Is there any estimates for the time it will take ...

**4**

votes

**1**answer

232 views

### Stationary, ergodic measures from the structuralist point of view

Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random ...

**5**

votes

**3**answers

457 views

### Ergodic action of a subgroup

Are there any examples of $H < G$ such that for any pmp ergodic action of the group $G$ on a standard proba space $(X,\mu)$ there exists a set $A$ of $\mu(A)>C$ such that the action of the ...

**0**

votes

**1**answer

218 views

### Entropy of inverse map for endomorphism case on surfaces

Hi,
I know that in the diffeomorphism case the measure entropy of the T:M^{2}-->M^{2} (M smooth Rimannian surface) will be the same as the measure entropy of T^{-1}. But i need to know about the ...

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votes

**2**answers

407 views

### proofs of ergodicity of Sl(2, Z) action on R^2 without using duality

The group $
G=SL(2, R)$ acts linearly on $\mathbb R^2$. The Lebesgue measure of $\mathbb R^2$ is invariant and ergodic for $G$. There is a proof using duality theorem:
Let $U$ be the upper triangular ...

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votes

**0**answers

98 views

### Is it a known example of adic transformation ? (2)

Please apologize: the Bratelli-Vershik graph in which I'm interested is not this one but this one:
At level $n$ there are $n$ vertices, there is one edge from each of the first $n-1$ vertices to the ...

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vote

**2**answers

354 views

### Is it a known example of adic transformation ? (1)

Here is a Bratelli-Vershik graph:
This graph should (I do not master this topic) define a Vershik adic transformation $T$ on its path space.
What are the invariant measure(s) on the path space ...

**6**

votes

**1**answer

253 views

### Ruelle inequality on a noncompact space

Does someone have a reference where the Ruelle inequality would be proved in the following context.
Let $M$ be a non compact smooth manifold, and $f:M\to M$ be a $C^1$-diffeomorphism (or $C^2$, ...

**6**

votes

**1**answer

383 views

### Aproximating dynamical systems by intrinsically ergodic systems

Let $X$ be a compact metric space and $f:X \to X$ a continuous map. We say that $(X,f)$ is approximated from below by a sequence of compact metric spaces $(X_i)_{i \geq 1}$ and a sequence of ...

**10**

votes

**2**answers

490 views

### Invariant measures and recurrent sets.

Suppose $T:X \to X$ is a homeomorphism of a compact metric space. The recurrent set is the set of all points $x \in X$ such that for every $\epsilon>0$ there exists an $n\in \mathbb{Z}$, $n \ne 0$, ...

**1**

vote

**0**answers

132 views

### Entropy of factors of Bernoulli schemes

Let $X$ be a Bernoulli scheme. A factor $\psi: X \to Y$ is finitary if for almost every $x \in X$ there exist integers $m \leq n$ such that the zero coordinates of $\psi(x)$ and $\psi(x')$ agree for ...

**1**

vote

**0**answers

191 views

### On the set of infinite measures

My question is about the structure of the set of infinite Borel measures on compact metric spaces invariant with respect to a homeomorphism.
Let $T$ be a homeomorphism of a compact metric space $X$ ...

**5**

votes

**0**answers

167 views

### Generating stationary, ergodic random fields on a homogeneous space

Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel ...

**4**

votes

**3**answers

326 views

### Equidistibution of horocycles through Hecke eigenvalues of Maass cusp forms

At the end of this very nice post:
http://blogs.ethz.ch/kowalski/2012/05/21/who-needled-buffon/
E. Kowalski talks about the equidistribution of the points $\frac{j+i}{N}$ when $j=1,\dots,N$ and $N$ ...

**2**

votes

**0**answers

170 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

202 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

178 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 ...

**8**

votes

**0**answers

202 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 ...

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vote

**0**answers

154 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

140 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

109 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

276 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

369 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 ...

**11**

votes

**3**answers

611 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

701 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

472 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 ...

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vote

**0**answers

214 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

304 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

76 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

210 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 ...

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vote

**0**answers

167 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, ...

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**0**answers

448 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 ...

**3**

votes

**2**answers

331 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

251 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

266 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 ...

**13**

votes

**5**answers

1k 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

389 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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vote

**0**answers

200 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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vote

**1**answer

414 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

660 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 ...

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votes

**1**answer

262 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 ...

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votes

**2**answers

236 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

206 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 ...

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votes

**1**answer

300 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 ...