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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5 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 ...
1
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0answers
19 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 $$ 1)\qquad\lim_{t\rightarrow\infty} \mu(\phi^{-t}(A)\cap B)=\mu(A)\mu(B); $$ $$ ...
1
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1answer
20 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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1answer
40 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
votes
1answer
58 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 ...
2
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0answers
25 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
29 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 ...
5
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0answers
125 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 ...
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0answers
25 views

Is mixing preserved under a measure theoretic-isomorphism of dynamical systems? [closed]

Show that mixing is an invariant of measure theoretic-isomorphism. I think that if I have two probability preserving transformation isomorphic together and one of them is mixing, I should prove ...
0
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1answer
27 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
39 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 ...
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0answers
28 views

Can ergodic theorem be used here

Suppose I have an ergodic Markov Chain $\{X_n\}$ where $X_n$ are bounded. Now, Can I say anything on the limit $$ \lim_{n\to\infty} \frac{1}{n}\ln E\left[e^{\sum_{i=0}^{n} X_i}\right]$$ I don't ...
1
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1answer
38 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
votes
1answer
34 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
votes
1answer
44 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
votes
1answer
46 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
votes
1answer
31 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
37 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, ...
1
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2answers
43 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
38 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
26 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
44 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. ...
1
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1answer
35 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 ...
2
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0answers
32 views

Geometric ergodicity and mixing - stationary case

I have this theorem: The Markov Chain {$X_n$} is stationary and geometrically ergodic if and only if {$X_n$} is stationary and absolutely regular with $\beta_n=O(\gamma^n)$ for some $\gamma\in ...
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0answers
60 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
68 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
votes
0answers
47 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 ...
1
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0answers
36 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
39 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
votes
1answer
154 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
votes
1answer
55 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} ...
2
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1answer
29 views

Question on Gauss map - application of Birkhoff's ergodic theorem

Take a Gauss map $G: [0,1] \longrightarrow [0,1]$ which is $$G(x) = \frac{1}{x} \mod 1, 0<x<1$$ and $0$ if $x=0$. Let $\mu$ be the Gauss measure. For $x \in [0,1]$ let $[a_{1}(x), ...
1
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0answers
23 views

Topological entropy, spanning sets and expansiveness of simple maps on a torus

I am trying to solve the following problem. Take the torus $\mathbb{T}^{2}$ and define the map $T(x,y)=(x + \alpha$ mod 1, $x+y$ mod $1)$, where $(x,y) \in [0,1]^{2}$. By induction, we have ...
2
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0answers
94 views

Conditional measure with respect to a sigma-algebra generated by the level sets of a function has full measure on its level set.

Let $(X,\mathscr{B},\mu,T)$ be a measure-preserving system, where $X$ is a compact metric space, $\mathscr{B}$ its Borel $\sigma$-algebra, $\mu$ a Borel probability measure and $T$ continuous. Let ...
0
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0answers
37 views

What connections between computer science and ergodic theory?

I have a background in ergodic theory, but I am switching to computer science/programming. I would like to know if tools from ergodic theory could be useful, especially something around coding of ...
0
votes
1answer
13 views

$\mathbf{Q}^*$-invariant measurable set of the real line

Let $X$ be some (Lebesgue-)measurable subset of $\mathbf{R}$ such that, for any rational $q \neq 0$, we have $qX=X$. Assume that the Lebesgue measure $\mu(X)$ of $X$ is $>0$. Does it hold then that ...
2
votes
1answer
42 views

What connections between machine learning and dynamical systems?

I have a background of ("pure") dynamical systems and ergodic theory, but I am switching to machine learning. Can some machine learning questions be treated from a dynamical systems/ergodic theory ...
0
votes
1answer
19 views

A problem on Markov process

Suppose, $\Pi_{\theta}$ be the transition probability function of a Markov chain. For any function $f$ define $$\Pi_{\theta}f_{\theta}(x) = \int f(y,\theta)\Pi_{\theta}(x,dy).$$ Is there any ...
1
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0answers
27 views

Application of Poincare recurrence to Baker's map?

Please see figures at http://www.mpipks-dresden.mpg.de/mpi-doc/kantzgruppe/wiki/projects/Recurrence.html. I heard that one of the applications of the Poincare recurrence theorem (which I do not ...
2
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0answers
22 views

On Ergodicity of the product of two Ergodic transformations

I was recently reading some textbooks and topics in Ergodic theory that I found this fascinating result which I couldn't prove completely: Suppose that $T$ and $S$ are two Ergodic transformations. ...
2
votes
1answer
42 views

necessity of uniform continuity for topological entropy

I am referring to Walters' book "Introduction to Ergodic Theory." When he defines the concept of topological entropy he always assumes that the transformation $T: X \rightarrow X$ is uniformly ...
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0answers
28 views

A question on Bernoulli measures and mixing

this is a question on ergodic theory. Suppose I have an integer $N \geq2$ and a probability space $(\sum^{+} , B, \mu_{p})$, where $\mu_{p}$ is the Bernouilli measure with respect to probability ...
2
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0answers
67 views

Birkhoff Ergodic Theorem Counterexample

I am trying to come up with a counterexample to this theorem under the assumption that the space is not sigma finite. I tried working with the power set of the real numbers with the measure $\mu(A) = ...
5
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1answer
58 views

A question on ergodic theory: topological mixing and invariant measures

This is a question on dynamical systems. Suppose I have a compact metric space $X$, with $([0,1], B, \mu)$ a probability space, with $B$ a (Borel) sigma algebra, and $\mu$ the probability measure. ...
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0answers
46 views

Ergodic Theory, Bernoulli Measure, Cylinder Set

Let $N\geq2$ be an integer and consider the probability space ($\Sigma^+$,B, $\mu_p$) where $\mu_p$ is the Bernoulli measure with respect to probability vector $\ p = (p_1,...,p_N)$ . Show that for ...
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0answers
52 views

Commutative Ergodic Theory

I've been working through some of the standard ergodic theorems for class (Birkhoff, Hopf, Chacon-Ornstein), and I'm attempting to extend Hopf's maximal ergodic theorem to the case of having ...
3
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0answers
38 views

Ergodicity and Appropiate Partition of the Space

I'm trying to solve the following problem. Let $(X, \mathcal{B}, \mu)$ be a probability space and let $T \colon X \to X$ be a measure preserving function. Prove that if $T^n \colon X \to X$ is ...
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0answers
26 views

Interpretation of a limit in ergodic theory

I have prove in a exercise that for a ergodic system $(X,\mathcal{B},\mu,T)$ if we take any integrable function $f$ with $\int_X f\,d\mu \neq 0$ then $$\lim_{N\to \infty} \log_N \left\vert ...
2
votes
0answers
55 views

Physical interpretation of Ergodicity.

If $R_{\alpha}:[0,1] \to [0,1]$ is defined by $$R_{\alpha}(x)=x+\alpha $$ then $R_{\alpha} $is called a circle rotation, and it is known that $R_{\alpha}$ is ergodic iff $\alpha$ is irrational. I ...
0
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0answers
44 views

Proving a theorem in Ergodicity

I read the theorem stated below on Wikipedia (http://en.wikipedia.org/wiki/Ergodicity#Formal_definition). But I do not understand how to prove the equivalence of these different definitions.Any hints ...