Question about probability spaces $(X,\mathcal B,\mu)$ with a measurable map $T\colon X\to X$ preserving the measure, that is $\mu(T^{—1}A)=\mu(A)$ for all $A$ measurable.

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1answer
37 views

Ergodic theory for flows: Invariant

Here is the definition of an invariant measure that I do know: Let $(X,\Sigma)$ be a measurable space and let $f\colon X\to X$ be measurable. A measure $\mu$ on $(X,\Sigma)$ is saif to be ...
3
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0answers
72 views

How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $ M ^ n $ be a Riemannian manifold, $ f: M \rightarrow M $ be ...
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1answer
46 views

What is an unique ergodic measure?

In a book I found the following: [...] on spaces of the form $\Omega\times \mathbb {R}^2$, where $\Omega $ carries a unique ergodic measure. What is meant with $\Omega$ carries a unique ...
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0answers
19 views

Invariant measure for 1-2x^2 and limit theorem

I have shown that $f(x) = 1-2x^2$ on [-1,1] has an invariant measure equivalent to lebesgue measure via the change of coordinates $h(x) = \sin\pi x/2$. (I.e. $g(x) = h^{-1}( f( h(x)))= 1 -2|x|$ has ...
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2answers
64 views

Conservative Measures under a group action (reference request)

I was reading a paper and the author define the concept of conservative measure: Let $(X,\mathcal{B})$ a measurable space and $G$ a group that acts on $X$ by $$G\times X:(g,x)\mapsto T_g(x)$$ where ...
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34 views

Stochastic matrix.

How do I show that a stochastic matrix, which is irreducible and at least one state is recurrent then all the other states are recurrent as well. And that the markov shift is conservative. The first ...
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3answers
177 views

Help with conditional expectation on the circle

Let $p >1$ a integer, $X = \mathbb{R} / \mathbb{Z}$ and $\mu\colon \mathcal{B}\to [0,1]$ a probability measure on the Borel subsets of $X$ which is $T \colon X \ni x \to (px \text{ mod }1)$ ...
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1answer
43 views

Limes superior of an ergodic and stationary process

Given a stationary and ergodic sequence of real-valued random variables $(X_n)_{n \geq 0}$ and a Borel$(\bar{\mathbb{R}})$-set A, with $\mathbb{P}[X_0 \in A] > 0$, I want to show, that ...
3
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1answer
55 views

Transformation preserving the Lebesgue measure

Let $f: [0,1] \to [0,1]$ a measurable transformation that preserves the Lebesgue measure $\mu$. So we have that $\liminf_n n\mid{f^n(x)-x}\mid\leq1$ $\mu$-almost everywhere. One trying to solve this ...
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0answers
19 views

Why are weak-mixing systems considered “random” and compact systems considered “ordered”?

As I understand it, weak-mixing systems sort of tend to become "orthogonal" to themselves on the long run, and compact systems tend to become almost periodic. How is this related to them being called ...
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1answer
52 views

Assertions about measures with computers

Let's consider the Lebesgue measure ($\mu$) over the closed interval $[0,1]$. As you know, $\mu(\mathbb{Q} \cap [0,1]) = 0$. In other way, as far as I know the computer just can represent accurately ...
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Question about B. Host paper 'Nombres, normaux entropie, translations'

I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out: Let $X = \mathbb{R}/\mathbb{Z}$, $p > 1$ a integer, $D_n = \{kp^{-n}\colon 0 \leq k < p^n ...
0
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1answer
100 views

Uniform Wiener-Wintner Theorem - proof

I am looking for proof of uniform version of Wiener-Wintner theorem: Let $(X, \mathcal{A}, \mu, T)$ be an ergodic measure preserving system. For $f \in L^1(\mu)$ which is orthogonal to the ...
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1answer
43 views

Operator induced by continuous function and measures

If $X$ is a compact metric space, and $T:X \rightarrow X$ is continuous map, what would be meant by $T_\ast$ is the operator on measures induced by $T$? Allow $\mu$ to be some Borel regular normed ...
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0answers
34 views

Ergodic Versus non-Ergodic Processes

Besides time averaging not carrying over to the ensemble average (in the limit), what are the pros and cons of ergodic and non-ergodic processes? Suppose you were in an engineering situation and you ...
1
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1answer
190 views

Not uniquely ergodic transformation

Could you teach me an example of NOT uniquely ergodic but ergodic transformation? And when any continuous, measurable, and ergodic transformation on a topological space X is uniquely ergodic, how ...
2
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1answer
76 views

Question about Pollicott-Yuri's proof of Rudolph theorem

On the book Dynamical System and Ergodic theory by Pollicott and Yuri there is a proof of the Rudolph x2 x3 theorem (page 153). It looks very clean in comparison with the original proof but I didn't ...
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2answers
87 views

Reference for Algebraic Groups in Ergodic Theory

It seems that the theory of algebraic groups is used in ergodic theory. I was hoping someone could recommend an introduction to algebraic groups that assumes a knowledge of commutative algebra ...
1
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1answer
33 views

Dynamical Systems Question on Definition

I'm working through some old notes on Dynamical systems, and I see a definition that I'm not familiar with. I'll call it property B, because I'm not sure what else to call it. In the notes, we assume ...
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0answers
29 views

Questions to the Maximal Ergodic Theorem

In Peter Walters' "An Introduction to Ergodic Theory" I found on page 37 Maximal Ergodic Theorem. Let $U\colon L_R^1(m)\to L_R^1(m)$ be a positive linear operator with $\lVert ...
2
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1answer
45 views

Questions to the proof of the $L^p$ Ergodic Theorem of Von Neumann

Before giving the Theorem of Von Neumann and asking my questions to its proof, I'll cite the Ergodic theorem of Birkhoff (out of Walters' "An Introduction to Ergodic Theory", p. 34) that is used in ...
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1answer
40 views

Invariant functions on product of ergodic systems is determined by eigenfunctions?

Given an ergodic measure-preserving system $(X,\mathcal{B},\mu,T)$, the product system $(X\times X, T \times T, \mu \times \mu )$ need not be ergodic, in other words: It may have non-trivial invariant ...
2
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0answers
25 views

Is a discrete random process issued from a sampled continuous ergodic WSS process also ergodic?

I have a continuous time process $\{X_t,t\in\mathbb{R}\}$ that is WSS and ergodic for the 1st and 2nd moments. I create a random discrete process $\{Y_n,n\in\mathbb{N}\}=\{X_t,t=nT\}$ by discretizing ...
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0answers
39 views

Mixing System and density argument

A Mixing system is defined as a dynamical system $(\Omega,\phi^t,\mu)$ for which the following relations holds $$ 1)\qquad\lim_{t\rightarrow\infty} \mu(\phi^{-t}(A)\cap B)=\mu(A)\mu(B); $$ $$ ...
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1answer
40 views

Conditions for “forward” measure-preservation

A transformation $T$ being $\mu$-invariant is by definition a transformation satisfying $$\mu(T^{-1} E) = \mu(E)$$ for all measurable sets $E$. I was wondering what are sufficient conditions for being ...
3
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2answers
73 views

Do Anosov flows exist on two dimensional compact manifolds?

Question: Let $M$ denote a two-dimensional, smooth, compact Riemannian manifold (without boundary). If $\phi:\mathbb{R}\times M \rightarrow M$ is a smooth flow, then prove that $\phi$ is not an ...
3
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1answer
80 views

Ergodic Theorem and flow

In Walters' An Introduction to Ergodic Theory on page 34 the Birkhoff Ergodic Theorem is given as follows: Suppose $T\colon (X,\mathfrak{B},m)\to (X,\mathfrak{B},m)$ is measure-preserving ...
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0answers
41 views

Why is $\left\{T^n x|n\geqslant 0\right\}$ dense in $X$ iff $x\in\bigcap_{n\geqslant 1}\bigcup_{k\geqslant 0}T^{-k}U_n$?

In Walters' An Introduction to Ergodic Theory I found the following Theorem and proof (p. 29): Theorem 1.7. Let $X$ be a compact metric space, $\mathfrak{B}(X)$ the $\sigma$-algebra of Borel ...
3
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0answers
40 views

A basic doubt on Markov chain/ergodicity

Consider a finite state (no. of state $N$) Markov chain $\{X_n\}$ (all the random variables are bounded) such that there is a state $i*$ such that $$ \sum_{i=1}^{N}p_{ii*}^{(n)} > 0$$ for all ...
6
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0answers
170 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with ...
0
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1answer
46 views

Ergodic (equivalent characterization)

Let $(\Omega,\mathcal{B},\mu,T)$ be a measuretheoretical dynamical system. Then this system is called ergodic if $$ B\in\mathcal{B}, T^{-1}(B)=B\implies \mu(B)=0\text{ or }\mu(B^C)=0. $$ ...
2
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2answers
58 views

Numerical integration using Birkhoff theorem

There is a method for numerical integration that uses Brikhoff ergodic theorem? For example if we have a irrational number $\alpha$ we know that for every continuous function $f \colon [0,1] \to ...
1
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1answer
72 views

Existence of ergodic joining

Let $\underline{X}=(X,\mathcal{B},\mu,T)$ and $\underline{Y}=(Y,\mathcal{B},\mu,S)$ be ergodic measure preserving systems on Borel probability spaces. A joining of $\underline{X}$ and $\underline{Y}$ ...
2
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1answer
38 views

Ergodic properties of orthogonal group $O(n)$

The orthogonal group $O(n)$ is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing ...
3
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1answer
69 views

Definition of strong mixing and definition of measure-preserving

I have a few questions regarding measure-preserving dynamical systems $(X,\mathcal{A},\mu,T)$. 1) The definition of measure preserving is always stated as $$\mu(T^{-1}B)=\mu(B),$$ for all $B$. I ...
2
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1answer
63 views

Unclear inequality in the proof of Birkhoff ergodic theorem.

I'm trying to understand the tricky proof of the ergodic theorem (Birkhoff 1931). My reference is "Ward,Einsiedler - Ergodic theory (with a view towards number Theory)" section 1.6: Consider the ...
2
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1answer
52 views

Obscure inequality in a passage of a proof (maximal ergodic theorem)

Look at the following excerpt from the book "Einsiedler and Ward- Ergodic Theory, with a view towards Number Theory": I don't understand why the inequality in the red box is valid. Maybe the ...
3
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0answers
132 views

Help understanding a geometric proof of the ergodicity of the Gauss measure for continued fractions

Any $x\in(0,1)$ can be written as a (regular) continued fraction $$ x = \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}} = [a_1,a_2,a_3,\dotsc] $$ An irrational number has a unique expansion, ...
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2answers
88 views

Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.
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2answers
88 views

Proof when the circle map is ergodic

Let $E=[0,1)$ with Lebesgue measure. For $a \in E$ consider the mapping $\theta_a:E \rightarrow E, \ \ \theta_a(x) = (x+a) \mod \ 1$. a) Show that $\theta_a$ is not ergodic when $a$ is rational. ...
0
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1answer
40 views

Proving that an ergodic and invariant map is constant a.e

I understand the first two sentences of the proof, however I cannot see how the third and final sentence holds. Why should $\mu(f \leq a)=0$ surely it should be non-zero as c is defined as the ...
0
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1answer
80 views

Using Kolmogorov's 0-1 law in proof of shift map being ergodic

Why should ${\cal E}_\theta$ be trivial?. I dont see how Kolmogorov's 0-1 law says that in this case we should take the 0 option. This is only mention of ${\cal E}_\theta$ in my notes I can find. ...
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1answer
57 views

For a non-compact metric space, do I have that the set of $\sigma$-invariant measures is compact?

Let $X$ be a non-compact metric space with a sub shift $\sigma: X \to X$. Do I have that the the space of $\sigma$-invariant probability measures on $X$ such that $\mu (B) = \mu (\sigma^{-1}(B))$ with ...
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0answers
68 views

Measure of intermediate local/pointwise dimension is not ergodic for $T_\beta(x)=\beta \cdot x \mod 1$ if $\beta \not \in \mathbb{N}$

Let $\mu$ be an ergodic measure for the map $T_a(x)=a \cdot x \mod 1$ for $a \in \mathbb{N}$ with $a \geq 2$ and invariant for $T_{a^{\frac{1}{n}}}$ for some $n \in \mathbb{N}$ (and thus also ...
2
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1answer
80 views

For a summable function, with summable variation, prove that $\sup_{i \in I} \sup_{x \in [i]}|f(x)| \exp((t-1) \sup_{x \in [i]}f(x) )$ is bounded

$\newcommand{\var}{\operatorname{var}}$ Let $X = \mathbb{N}^\mathbb{N}$ and $f: X \to \mathbb{R}$ be a function such that $$|f|_{\var} = \sum_{i=1}^{\infty} \var_n f < \infty,$$ where $\var_n f = ...
3
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0answers
84 views

Strange definition of Ergodicity

In an engineering course, a stationary process was defined to be ergodic if for all $k\in \mathbb{N}$ and for any bounded (measurable) function of $k+1$ variables we have $$\lim_{N\rightarrow ...
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0answers
41 views

Is a pot of boiling water an example of non-ergodic process?

Sorry if this question is a bit dumb... I think (but correct me if I'm wrong) that ice cream moving in a perfect ice cream maker is an example of ergodic flow: the flow itself is conserved, no ...
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0answers
49 views

Is Kolmogorov's zero–one law undecidable?

Kolmogorov's zero–one law is related to other parts of probability like the law of large numbers. However it is stated that what the actual probability of a tail event is (either 0 or 1) is hard to ...
5
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1answer
176 views

Entropy of the induced transformation

I need help with this problem: Let $(X,\mathcal{B},\mu,T)$ be a ergodic dynamical system in the probability space $(X,\mathcal{B},\mu)$. Let $A \in \mathcal{B}$ with $\mu(A)>0$. We define the ...
3
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1answer
62 views

Correlation Sequences and Unitary Operators

Let $U:H \to H$ be an unitary operator on a Hilbert space $H$. Suppose that $x \in H$ is orthogonal to all the eigenvectors of $U$. I'd like to prove that $$ \lim_{N \to \infty} \frac{1}{N} ...