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

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votes

**3**answers

226 views

### The property of a Markov measure

Given $\sigma$ a shift map, $m$ - a Markov measure, $C_a$, $C_b$ - cylinder sets.
Suppose $P \in C_b$. The problem is to show the following
\begin{equation}
m(C_a \cap \sigma^{-1}(P)) = \frac{m(C_a ...

**5**

votes

**1**answer

329 views

### Non-existence of ergodic measures

Good afternoon.
Can anybody give me an example of a continuous map $T:X\to X$ defined on a Polish space $X$ which admits an invariant Borel probability measure but no ergodic Borel probability ...

**1**

vote

**1**answer

161 views

### power bounded adjacency matrices

A bounded linear operator $T$ on a Banach space $X$ is called power bounded if $\|T^k\|\le M$ for some $M>0$ and all $k\in \mathbb N$.
A classical result of Lorch says that if $X$ is reflexive, ...

**4**

votes

**1**answer

273 views

### Characterising ergodicity of continuous maps

Hello all.
Suppose $X$ is a Polish space, $\mu$ is a Borel probability measure on $X$, and $T:X \to X$ is a continuous $\mu$-preserving map which is not ergodic.
Does there necessarily exist a Borel ...

**0**

votes

**1**answer

168 views

### On the affine property of entropy map

Theorem 8.1 in Wlaters' book "An introduction to ergodic theory" says the entropy map is affine. Namely, let $T:X\to X$ be a continuous map of a compact metrc sace. If $\mu, m\in M(X,T)$ and ...

**0**

votes

**1**answer

119 views

### Ergodicity with respect to the shift

On the space $S=\{ 0,1,\ldots,m \}^{\mathbb{N}}$ for some $m\in \mathbb{Z}_{+}.$ And given a probability $\mu$ on it. Is it true that $\mu$ is fully supported if and only if it is ergodic for the ...

**3**

votes

**1**answer

275 views

### Invariant measures for Cellular automata

An easy question that I have never been able to answer.
Suppose we have the CA on $\{ 0,1,2 \}^{\mathbb{N}}$ with local rule given by $f(x,y)=A_{x,y}$ and $A$ the $3\times 3$ matrix ...

**5**

votes

**2**answers

239 views

### pointwise ergodic theorem and mean sojourn time

Originally posted on Maths StackExchange, but repositing here because of getting no answer there. Not a research question really - I'm just confused by implications between various ergodic ...

**6**

votes

**0**answers

238 views

### Do ergodic isometries have discrete spectrum?

Let $X$ be a metric space, $\mu$ a Borel probability measure, and
$T:X\rightarrow X$ be an ergodic measure preserving isometry.
Is $(X,\mu,T)$ measure theoretically isomorphic to a minimal isometry ...

**2**

votes

**0**answers

159 views

### Partitions of central sets via dynamical systems

In the book "Recurrence in Ergodic Theory and Combinatorial Number Theory", 1981, Furstenberg introduced the notion of central sets.
He proved in Theorem 8.8 that in each finite partition of ...

**6**

votes

**7**answers

680 views

### Quantization of a classical system (e.g. the case of a billard)

There has been already several questions asking for an introduction to quantum mechanics
for a mathematician, but this ons is slightly different, and more restrictive. I know (some)
quantum mechanics, ...

**5**

votes

**0**answers

182 views

### Fibre Mixing for Dynamical Systems

Hi all,
I'm interested in understanding a fairly difficult theorem of Lindenstrauss Peres and Schlag. In that paper the authors prove that certain dynamical systems related to beta expansions and ...

**5**

votes

**1**answer

233 views

### Failure of the Pointwise Ergodic Theorem

It is known that Birkhoff's pointwise ergodic theorem (unlike von Neumann's mean ergodic Theorem) fails to hold for general Folner sequences.
The counter-example usually given is the Folner sequence ...

**1**

vote

**2**answers

129 views

### Relation between entropy of one-parameter group and single elements of this group

My question is motivated by the hypothesis of the Lindenstrauss' proof of arithmetic quantum unique ergodicity, and the answer to my question is certainly known. However, I could not find it in the ...

**4**

votes

**2**answers

339 views

### Question about entropy

I see this question on the math stack exchange. I found it interesting and still there is no solution there
Let $(X,A,\nu)$ be a probability space and $T:X\to X$ a measure preserving ...

**9**

votes

**3**answers

556 views

### Examples In Ergodic Theory and Topological Dynamics

I am currently studying basic Ergodic Theory:
Invariant Measures
Poincaré recurrence Theorem
Invariant Measure For Continuous Transformations
The Ergodic Theorems and Applications
Ergodic ...

**6**

votes

**2**answers

247 views

### Minimal period of arithmetic progressions occurring in sets of positive density.

Let $A$ be a subset of ${\mathbb N}$ with positive upper-Banach density, and for each integer $k\geq3$, define $R_k=R_k(A)$ to be the smallest positive integer $r$ such that $A$ contains a length $k$ ...

**3**

votes

**2**answers

153 views

### Estimate entropy of a binary process in terms of decay of correlations

Suppose $( X_{n} )$ is an ergodic binary process with
$$
\mathbb P(X_{n}=1)= \mathbb P(X_{n}=0)=\frac 12.
$$
Naturally the entropy (rate) $h(X)$ of $X=(X_{n})$ satisfies
$$
h(X)=\lim_{n\to\infty} ...

**4**

votes

**1**answer

185 views

### Practical way to check for geometric convergence

Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution.
When I measure the convergence rate ...

**3**

votes

**3**answers

598 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

195 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

303 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

368 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

278 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

346 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?

**9**

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**2**answers

315 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

254 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

496 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

220 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 ...

**8**

votes

**2**answers

438 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 ...

**0**

votes

**0**answers

100 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 ...

**1**

vote

**2**answers

393 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

279 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

400 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

587 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

138 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

206 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 $X$ ...

**6**

votes

**0**answers

187 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

373 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

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 ...

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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

234 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

222 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

159 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

143 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 ...

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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

280 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

443 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

653 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

746 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 ...