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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2
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1answer
37 views

Continuity in the Krylov-Bogoliubov theorem

I have the following proof of the Krylov-Bogoliubov theorem, which asserts that given a compact metric space $X$ endowed with a continuous transformation $T \colon X \to X$ one can find a ...
1
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0answers
64 views

Rate of convergence of $n$-fold convolution over $\mathbb{Z}_p$.

Suppose that $\mu$ is a probabilist measure on $\mathbb{Z}_p$ such that $d_{TV}(\mu,\mu_U)\leqslant \delta < 1$. What are the best upper bound on $$ d_{TV}(\underbrace{\mu*\mu*\ldots*\mu}_{n ...
2
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1answer
71 views

Unique Ergodic Measure

From <http://mathworld.wolfram.com/ErgodicMeasure.html> "If there is only one ergodic measure, then $T$ is called uniquely ergodic. An example of a uniquely ergodic transformation is the map ...
1
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1answer
45 views

compactness of the set of invariant measures

Suppose that we have a dynamical system on some compact space $X$ with discrete time space and transformation given by some $\phi : X \rightarrow X$. My question is when is the set $Prob(\phi)$ of all ...
3
votes
1answer
42 views

If $f^n$ is mixing then $f$ is mixing?

Let $(X,\mathcal{A},\mu)$ be a probability space and $f:X\to X$ be a measurable map that preserves $\mu$. Fix $n\in \mathbb{Z}^+$. It's not hard to see that $f$ ergodic does not necessarily imply ...
2
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0answers
56 views

Generalizing the ptwise or $L^1$ ergodic theorem

I was able to write a proof based on the hardy-littlewood maximal function of the following statement: Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space, and $d=n \geq 1$ be fixed. Let ...
2
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0answers
70 views

simple proof of the $L^2$ weak law for discrete-time ergodic Markov processes

Let $\{X_t\}_{t\in\mathbb{Z}}$ be a stationary and ergodic stochastic process with finite second moment. Von Neuman's ergodic theorem implies that the time average $(1/N)\sum_{j=0}^{N-1} X_j$ ...
1
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1answer
93 views

Poincare recurrence theorem and convergence on compact metric spaces

I am looking for a proof of the following theorem: Let $X$ be a compact metric space with metric $d$, endow $X$ with the Borel $\sigma$-algebra and a probability measure $\mu$. Let $T\colon X\to X$ ...
3
votes
2answers
92 views

Ergodic for the mean, but not ergodic stochastic process?

Is there an example of a strictly stationary (zero mean, finite variance) stochastic process $(X_t\mid t\in \mathbb{N})$ that satisfies the conclusion of the ergodic theorem, i.e., the sample mean ...
8
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1answer
118 views

An ergodic theorem on the circle

Let $S^1$ be a circle (i.e. a closed $1$-dim. manifold) and let $F$ be a non-vanishing smooth vector field on $S^1$. Denote by $(t,x) \mapsto \Phi_t^x$ the flow generated by $F$. I want to show ...
3
votes
1answer
44 views

If $T^m$ is ergodic, so is $T^{m^2}$?

The HW problem: If $T^m$ is ergodic, show $T^{m^2}$ is ergodic. (Where we can assume $T$ is measure-preserving transformation on a probability space, I think. It wasn't in the problem, but everything ...
4
votes
1answer
93 views

What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
0
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1answer
163 views

How does chaos arise in Hamiltonian systems?

I have a question about how chaos arises in Hamiltonian systems. I've been taking a upper year undergrad class on dynamics and it has focused on chaos a lot. So far however we have only seen the more ...
3
votes
0answers
50 views

Ergodic mean for Schrodinger flow

Let us consider the linear Schrodinger equation in $\mathbb{R}^N$ $$ (i\partial _t+\Delta)\,u=0\mbox{ ,}\quad u(0,x)=f$$ with $f\in L^2(\mathbb{R}^N)$, and let $u(t,x)=e^{it\Delta}f$ its solution. ...
4
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1answer
65 views

It Suffices to Check Mixing on an Algebra

Let $X, \mathcal{A}, \mu$ be a probability space and $T: X \rightarrow X$ a measure preserving measurable map (i.e. $\mu (T^{-1}(A)) = \mu (A)$ for all $A \in \mathcal{A}$). We say $T$ is mixing for ...
2
votes
1answer
38 views

Spectrum and tower decomposition

I'm trying to read "Partitions of Lebesgue space in trajectories defined by ergodic automorphisms" by Belinskaya (1968). In the beginning of the proof of theorem 2.7, the author considers an ergodic ...
4
votes
1answer
241 views

The Denjoy Theorem

I'm currently studying Denjoy's theorem, which says the following: Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $ \rho$ and the variation of $f^{'}$ (denoted by ...
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1answer
57 views

A random variable, measurable w.r.t. the class of invariant events, is invariant

I would like to deeply understand the following proof; however, many things are not really obvious to me. I would appreciate if somebody could explain the solution in detail. Thank you. Problem ...
2
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1answer
68 views

Ergodicity in Transformation Implied by Ergodicity in Induced Transformation

For a finite, recurrant, invertible, measure preserving dynamical system with transformation $T$, I can show that if $T$ is ergodic, then the induced transformation for any positive-measure set is ...
2
votes
1answer
108 views

Proving Borel Cantelli Lemma using Martingales

I need a hint for exercise 5.2.1 in the book: Ergodic Theory: with a view towards Number Theory By Manfred Leopold Einsiedler, Thomas Ward. In the chapter 5 the authors gives the margingale ...
3
votes
1answer
103 views

$\mu(E)>0 \Rightarrow \mu(f(E))=\mu(E)$?

Consider injective continuous functions $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. Which of these function are such that: $\mu(E)>0 \Rightarrow \mu(f(E))=\mu(E)$? (Isometries are clearly ok.)
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0answers
71 views

Different definitions of an ergodic stationary process

From page 3 of a note: A stationary process is ergodic if any two variables positioned far apart in the sequence are almost independently distributed. A formal definition is the following: ...
1
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1answer
51 views

Wandering Sets and Recurrent Transformations

I'd like to show that a measure preserving transformation $T:X\rightarrow X$ is recurrent iff it has no wandering sets of positive measure. I'm working from the following definitions: a measurable ...
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0answers
32 views

Recurrence for upper Banach Density

Let $E\subseteq \mathbb{Z}$ have $d^*(E) >0$ where $d^*$ is upper Banach density. I am trying to understand why there must exist an $1\leq n \leq \frac{1}{d^*(E)} + 1$ such that $d^*(E\cap (E-n)) ...
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2answers
59 views

a question on orbit in ergodic theory

For the map $T: [0,1]\to [0, 1]$ defined by $Tx=10x\pmod{1}$, how to use the decimal expansion to construct a $x$, such that the orbit of $x$, say $\theta_x=\{T^nx: n\geq 0\}$ is dense in $[0, 1]$ ...
3
votes
3answers
298 views

Relationship between entropy and ergodicity

Are there any direct connections between entropy and ergodicity? For example, does knowing that $(X,\mathcal{S},\mu,T)$ is ergodic help in computing the entropy? I know that there are some indirect ...
4
votes
1answer
242 views

Liouville's theorem: How to get an invariant measure?

Liouville's theorem states that the `natural' 2-from is preserved under the Hamiltonian flow. Apparently this leads to an invariant measure $\mu$ as follows \begin{equation} d\mu = \frac{d\sigma}{|| ...
3
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0answers
107 views

Reference for a transformation

Has the (Lebesgue-)ergodic transformation $T: \{0,1\}^{\mathbb{N}} \to \{0,1\}^{\mathbb{N}}$ defined by $T(x(0)x(1)x(2)\cdots) = x(1)x(3)x(5)\cdots$ been well-studied? If so, where? Does it have a ...
0
votes
1answer
56 views

Correspondence between fractal sets and trees

In Hillel Furstenberg's series lectures on ergodic theory in fractal geometry, he mentioned his search on finding a one-to-one correspondence between fractal sets and trees, however, I couldn't not ...
6
votes
1answer
421 views

Ergodic action of a group

What does it mean and how is it defined if the action of a group is meant to be ergodic? Thank you for your replies!
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0answers
32 views

literature to learn more on ergodic harris recurrent chains with an atom

I'm trying to learn more on the topic mentioned in the title. Namely I'd like to get more information on the behavior of the boundary terms. ie if I decompose sum of my chain (suppose it's ...
10
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1answer
219 views

Non-ergodic measure

Is there an easy way to see that if $\mu$ and $\nu$ are $T$-invariant measures on the same space $X$, and $\mu \neq \nu$, then $\frac{1}{2}(\mu+\nu)$ is NOT ergodic? I know that ergodic measures are ...
2
votes
1answer
73 views

Eigenfunction of $T^n$ vs. eigenfunction of $T$ when $T$ is ergodic

Let $(X, \mathcal{B}, \mu, T)$ be an ergodic system, and suppose $f \in L^2(X, \mathcal{B}, \mu)$ is an eigenfunction of $T^n$ for some integer $n > 1$. Is it possible to write $f$ as a linear ...
4
votes
2answers
93 views

Optimal probability measure

Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
2
votes
0answers
71 views

What is the name of this metric: Why is $(\mathcal{M}, L)$ complete

I am reading section 4 of this article about invariant measures: http://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf Let $(X,d)$ a complete metric space, ...
3
votes
2answers
144 views

Ergodic theory question about the support of a measure.

I am trying to work through some problems from " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question: ...
2
votes
1answer
219 views

Are polynomials modulo $1$ equidistributed?

It is an elementary exercise to show that for irrational $\alpha \in \mathbb{R}$, the sequence $\{ \alpha n \mod 1 \}_{n \in \mathbb{N}}$ is dense in $T := \mathbb{R}/\mathbb{Z}$. With more work, it ...
6
votes
2answers
140 views

Recurrent point of continuous transformation in a compact metric space

Given a metric space $(X,d)$ and a transformation $T:X\rightarrow X$, a point $x\in X$ is said to be recurrent iff it belongs to the closure of its orbit $\{T(x), T^2(x),...\}$: more precisely, ...
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0answers
60 views

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
64 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
votes
0answers
93 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
votes
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 ...
7
votes
2answers
206 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 ...
2
votes
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 ...
0
votes
1answer
171 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.
2
votes
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
votes
0answers
128 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 ...
2
votes
0answers
88 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?