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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convergence of discrete random variables with finite entropy

Let $Z$ be the set of discrete random variables on some probability space. Define the quantity $d(X_1,X_2)=h(X_1 \mid X_2)+h(X_2 \mid X_1)$ between two random variables $X_1, X_2 \in Z$. For $X \in Z$ ...
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0answers
69 views

Furstenberg recurrence

I am having trouble verifying exercise 1 here. Can I get some hints/solutions? (It's not homework, I am just reading up on it for my own interest.) Theorem 2. (Furstenberg multiple recurrence ...
3
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0answers
96 views

Are geodesic flows on surfaces with negative curvature Anosov?

I'm just going through the original book by Anosov, where he tries to proof this result. I don't quite understand it. So let $\phi_t:TM\rightarrow TM$ be a geodesic flow on a compact surface $M$ of ...
2
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0answers
74 views

The chacon transform

I am following this document http://www.jstor.org/stable/2037431?seq=4 Shouldn't it be necessary to check that the chacon transform is ergodic? The theorem I'm familiar is this: Let $T$ be a ...
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2answers
207 views

What is the distribution of leading digits of the squares?

Inspired by How can 0.149162536... be normal?, I ask for the distribution of the leading coefficients of $1,4,9,16,25,36\ldots$. (namely $1,4,9,1,2,3\ldots$) Does a Benford-like law apply? The online ...
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1answer
62 views

Why every strict stationary process have the following representation

Suppose $\{X_n\}_{n\geq 1}$ is a strict stationary process, that is for every $n$, $(X_0,\ldots,X_n)$ and $(X_1,\ldots,X_{n+1})$ have the same distribution. Then there is a probability space ...
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1answer
175 views

Unfolding a Billiard Trajectory

The following image is from page 1019 of http://www-fourier.ujf-grenoble.fr/~lanneau/references/masur_tabachnikov_chap13.pdf From what I understand about unfolding billiards we are representing the ...
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0answers
53 views

Invariance of conservative part in Hopf decomposition

Let $(\Omega,\mu)$ be a $\sigma$-finite measured space. Let $\tau$ be an endomorphism of this space, meaning that $\mu(\tau^{-1}(A))=\mu(A)$ for any $A$. There is a decomposition $\Omega=C \cup D$ ...
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1answer
39 views

Need help with the following:

Proof or counterexample: a) if $T$ is ergodic, then $T^2$ is ergodic, b) if $T$ is strong mixing, then $T^2$ is strong mixing. Thank you.
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2answers
75 views

A question involving Invariant Set in ergodic theorem

I have a question about the invariant set in the ergodic theorem, I am wondering if anyone could give me some help or hint. In the measurable space (X, $\Sigma$) and consider a measurable self map T, ...
2
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0answers
129 views

Bernoulli shift on $S^\mathbb{Z}$

Why is the Bernoulli shift ergodic? I know the proof for $S^\mathbb{N}$. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from ...
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0answers
91 views

Ergodicity and equidistribution

It is known that ergodicity imply dense, but not vice-versa. Also dense does not imply equidistribution (example). But what about equidistribution and ergodicity properties?
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1answer
224 views

Ergodic Theory (Weak Mixing)

If $T$ is weak mixing then we know that $T\times T \times \ldots \times T$ is also weak mixing. Does anyone know if this is true for $T\times T \times \ldots$?
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1answer
73 views

Recipe to proof ergodicity?

Consider the following theorem about equivalent formulations of ergodicity. Let $S$ be a measure preserving map on a measrespace $(\Omega,\mathfrak F,\mathbb P)$ and define $$\nu_n(A,\omega) = ...
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5answers
271 views

High-School Level Introduction to Dynamical Systems

In one month I'll be giving a talk to motivated high schools students on a topic of my choice from dynamical systems and/or ergodic theory. I'm having trouble coming up with a topic compelling enough ...
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1answer
224 views

Why is integer approximation of a function interesting?

I have recently learnt the following result: Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is ...
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1answer
119 views

An application of Birkhoff ergodic theorem

Let $(X,B,m)$ be the particular probability space where $X$ is the circle in the plane with center at the origin and radius 1, $B$ is the collection of borel sets, $m$ is the lebesgue measure. Let ...
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1answer
181 views

A necessary and sufficient condition for ergodicity

Let $(X,\mathcal B,\mu)$ be a probability space and $T\colon X\rightarrow X$ be measure preserving. I need to prove that $T$ is ergodic if and only if the following property holds: If $f\colon ...
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0answers
79 views

Checking isomorphism between two measure preserving maps

Let $T_1\colon [0,1]\rightarrow [0,1]$ be the full tent map, $B_1$ be the collection of Borel sets, and $m_1$ denote the Lebesgue measure. Let $T_2\colon\sum_2^+\rightarrow \sum_2^+$ be the full one ...
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1answer
43 views

Historical behavior of the Birkhoff averages

Birkoff Ergodic Theorem: Let be $(X,\mathcal{A},\mu)$ a probability space, $T:X\to X$ a measurable tranformation preserving $\mu$ and $f\in L^{1}(\mu)$, then there exists a $\Sigma \subset X$ and ...
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1answer
44 views

Need help in proving a set has bounded gaps.

Let $(X,B,\mu)$ be a probability space and $T\colon X\rightarrow X$ is measure preserving. Let $A\in B$ such that $\mu(A)>0$. Then I am asked to prove the following claims: The set of positive ...
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0answers
23 views

Need help in proving a unique invariant borel probability measure is ergodic. [duplicate]

Let $X$ be a topological space and $f : X \rightarrow X$ be a function. Suppose that there exists a unique invariant borel probability measure m. We need show that m is ergodic : I assumed by ...
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1answer
111 views

Prove that $m$ is ergodic.

Let $X$ be a topological space, $f\colon X\rightarrow X$ be a function. Suppose that there exists a unique invariant Borel probability measure $m$. Prove that $m$ is ergodic. Thank you.
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71 views

Conclusions about the Manneville-Pomeau Example

I was trying to solve an exercise about the Manneville-Pomeau example, but I got stuck. I need to give a little background before posting the exercise. Thank you guys in advance for your attention. ...
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1answer
87 views

Finding invariant Borel probability measures for a contraction map

Let $X$ be a compact metric space. Let $f:X\rightarrow X$ be a contraction map. I need to find all $f$-invariant Borel probability measures. Thank you.
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1answer
191 views

Measurable Partition and Ergodic Decomposition

I need some background before asking the question: Let $\mathcal{P}$ is called a measurable partition if there is a measurable set $M_0\subset M$ with full probability measure such that, restric to ...
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1answer
32 views

Finding a minimal non-empty closed $G$-invariant set of a compact metric space

Let $G$ the abelian group generated by commuting homeomorphisms $f_1,\dots,f_q:M\rightarrow M$, where $M$ is a compact metric space. Show that there is $X\subset M$ minimal with respect to the ...
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0answers
28 views

When does a flow inherit ergodicity from a Poincare section?

Assume we have a smooth compact manifold $M$ with boundary and a smooth complete vector field $X$ on $M$. Let $\phi^{t}$ be the resulting flow and let $\mu$ be a probability measure on $M$. Define the ...
3
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1answer
74 views

pointwise ergodic theorem and mean sojourn time

Let $G$ be a group and let $F_i$ be a sequence of finite subsets of $G$. Suppose $G$ acts on a probability measure space $(X,\mu)$ in a measure preserving way, and suppose that this action is ergodic. ...
2
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1answer
128 views

stationary and ergodic

Let $p\in \mathbb N$ and $a_1,\ldots,a_p\in \mathbb R $. Denote by $x$ the sequence $$x=(x_k)_{k\in\mathbb N }=(a_{k \bmod p})_{k\in\mathbb N }$$ where $(k \bmod p)\in{1,\ldots,p}$ is the remainder ...
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1answer
77 views

How to check the strong ergodicity of the $SL_2(\mathbb{Z})$-action on the torus?

Suppose $\Gamma\subset SL_2(\mathbb{Z})$ is a non-amenable subgroup, especially, $\Gamma=SL_2(\mathbb{Z})$. Consider the natural action of $\Gamma$ on $S^1\times S^1=T^2$. How to check that this ...
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2answers
61 views

Convergence of an Ergodic process

I'm having trouble working through the math of the problem below. I believe the problem as described is an ergodic process. I've written a simple simulation of the problem, that converges to 66.6...6% ...
3
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1answer
119 views

convergence of entropy and sigma-fields

This question is related to this one. Let $(X_1, X_2, \ldots)$ be a sequence of random variables such that each $X_n$ takes its values in a finite space, say $\{0,1\}$, and the $\sigma$-field ...
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2answers
220 views

Von Neumann's ergodic theorem

Where can I find the proof of Von Neumann's ergodic theorem? Please, give me references or write names of books where I can find it.
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2answers
97 views

Can the time mean over a dense orbit equal the space mean for arbitrary functions?

Let $\varphi : M \to M$ be a measure-preserving map of a measure space $M$ with measure $\mu$, and let $f \in L^1(\mu)$ be arbitrary. If $p$ is the starting point of an orbit that is dense in $M$, ...
2
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1answer
199 views

Ergodicity of Geodesic Flow

I know the Birkhoff Ergodic theorem; and I know what is a Riemannian manifold and what a geodesic is. I also read the definition of geodesic flow on the tangent bundle of one such. But I do not yet ...
3
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1answer
126 views

Where does ergodic come from?

In math you usually understand why terms such as triangle, function, polynomial, category or even vector. However where does the word ergodic come from? Does it have a meaning in another language? ...
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1answer
95 views

Lyapunov Exponent

Suppose $(X,A,\mu)$ a probability space, where $X$ is a compact Riemann manifold, $T:X\to X$ a diffeomorphism and $T$ is a measure-preserving transformation( over the borel $\sigma$ algebra). Prove ...
2
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2answers
76 views

Exposition about Ergodic Entropy

does anybody could suggest me any book or paper about Entropy in Ergodic Theory? I'm trying to prepare an exposition but I've just 30/40 minutes more or less so I'd like to choose a theme, or some ...
5
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1answer
253 views

Pinsker $\sigma$ Algebra

Let $(X,A,\nu)$ be a probability space and $T:X\to X$ a measure-preserving transformation. The Pinsker sigma algebra is defined as the lower sigma algebra that contains all partition P of measurable ...
4
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1answer
154 views

About Ergodic Theorems

Is there any demonstration purely dynamic of the Birkhoff Ergodic Theorem, i.e, without the Maximal Ergodic Theorem?? I ask this question because I never understood the intuition behind The Maximal ...
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1answer
219 views

The prime numbers do not satisfies Benford's law

A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability, $$ P(d)=\log_{10}(d+1)-\log_{10}d,$$ how do you prove that the prime numbers do ...
2
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1answer
144 views

Question about Benford's law

A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability, $$ P(d)=\log_{10}(d+1)-\log_{10}d$$ With Birkhoff's Ergodic Theorem is possible ...
3
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1answer
143 views

Question about entropy

Let $(X,A,\nu)$ be a probability space and $T\colon X\to X$ a measure preserving transformation $\nu$. Take a measurable partition $P=\{P_0,\dots,P_{k-1}\}$. Let$I$ be a set of all possible ...
3
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1answer
159 views

When is the composition operator assigned to a measure-preserving map unitary?

Let $(X,\mathcal{B},\mu)$ be a standard probability space, and let $T:X\rightarrow X$ be a measurable, measure-preserving transformation, i.e. for every $A\in\mathcal{B}$, $\mu(T^{-1}(A))=\mu(A)$. ...
4
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1answer
535 views

Why are hyperbolic toral automorphisms (e.g. Arnold's cat map) ergodic?

Let $\varphi : \mathbb{T}^2 \to \mathbb{T}^2$ be a hyperbolic automorphism of the torus, induced by a linear map $A : \mathbb{R}^2 \to \mathbb{R}^2$ of determinant $\pm 1$ with no eigenvalues of ...
14
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1answer
468 views

Kakutani skyscraper is infinite

Karl E. Petersen's book "Ergodic Theory", chapter 2, exercise 9, on page 56 Prove that for any ergodic measure preserving transformation $T:X\rightarrow X$ on a non-atomic probability space $(X, ...
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1answer
34 views

How to formulate an integral on the probability space of a Markov shift?

For a problem I am currently working on, I would like to calculate the integral over the space $(\{1,2,\dots,n\}^\mathbb{N},\mathcal{F},\nu)$ where $\nu$ is the markov measure. My problem is that ...
11
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1answer
166 views

Square root of an integer has only even digits

Is there a non-square positive integer $n$, that $\sqrt{n}$ has only even digits in its decimal representation ?
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1answer
96 views

How does the Markov measure extend to cylinders with 1 specified coordinate?

Consider a Markov Chain $(X,\mathcal{F},\mu,T)$, where $X = (1,2,\dots,n)^\mathbb{Z}$ and $T$ is the left shift and a transition matrix $P=(p_{i,j})$ and stationary distribution $\pi$ such that $\pi P ...