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Questions tagged [ergodic-theory]

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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Definition of ergodicity and ergodic process

I am confused by the definitions of ergodicity in wikipedia, see formal definition here which says that a measure-preserving transformation $T$ is ergodic if for every event $E$, $T^{-1}(E) = E$ ...
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Under a measure preserving transformation of a probability space, I need to show the following.

Given a measure preserving system (X,$\mathcal{B}$, $\mu$, T) where $\mu$ is a probability measure, I want to show that $\forall$ A $\in\mathcal{B}$ and $\forall \epsilon >$ 0, $\exists$ n $\in\...
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Koopman-von Neumann Decomposition Properties

Sorry if the title of this question is vague--I'm open to suggestions. For this question, we're working in a probability space $(X, \mathcal{M}, \mu)$. In a proof of "ergodic Roth's theorem" given ...
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Gauss measure and continued fraction

For $x \in [0,1)$ then the continued fraction representation of $$x=0 + \cfrac{1}{a_1(x)+\cfrac{1}{a_2(x)+\cfrac{1}{a_3(x)+\cfrac{1}{\dots}}}}$$ can be written as $[0; a_1(x), a_2(x), a_3(x), \dotsc]$ ...
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Does the property of uniquely ergodic imply that the map has a unique chaotic attractor for all $c$ in $D$

Definition: Let $(X,B)$ be a measurable space and let $T:X→X$ be a measurable transformation. If there is a unique $T$-invariant probability measure then we say that $T$ is uniquely ergodic. Consider ...
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Problem in Ergodic theory

Let $(X,T,\mu)$ be a classical dynamical system, where $(X,\mu)$ is a probability measure space and $T$ is a measure preserving invertible transformation. Let $U$ be the unitary on $L^{2}(X,\mu)$ ...
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Unique ergodicity and first return time

I'm trying to solve the following problem: Let $T\colon X\to X$ be a continuous map on a compact metric space $X$, uniquely ergodic. Let $Y\neq \emptyset$ be an open set. Show that $t(x) = \min\left\{...
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$d(\xi, \eta) = H_{\mu}(\xi|\eta) + H_{\mu}(\eta|\xi)$ defines a metric

I want to show that $d(\xi, \eta) = H_{\mu}(\xi|\eta) + H_{\mu}(\eta|\xi)$ defines a metric on the space of all partitions (considered up to sets of measure zero) of a probability space $(X, \mathscr{...
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About topological conjugacy

The Smale horseshoe map $f$ is desribed in this page: What's the point of a Horseshoe map? A striking feature of this system is the stability of its dynamics: given any diffeomorphism $g$ ...
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Von Neumann ergodic theorem for purely continuous spectrum

$\newcommand{\1}{1\negthickspace{\mathrm{I}}}$ Von Neumann ergodic theorem states that, if $U(t)$ is a one-parameter group of unitaries acting on a Hilbert space $\mathcal{H}$, we have $$ \lim_{T\to ...
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Understanding $T(x)=3x\pmod 1$ [closed]

Let $T:[0,1]\to[0,1]$ be such that $T(x)=3x\pmod{1}$ which is measurable with respect to the $\sigma$-algebra of Borel on $[0,1]$, which we denote by $\mathscr{B}_{[0,1]}$. Prove that Lebesgue ...
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Could anyone help me to prove the followings are equivalent?

$K\subseteq \mathbb R^n$ be closed then show that TFAE A probability measure $\mu$ on $K$ is ergodic. (1) Every $\mu$-invariant set of $\mu$ measure zero or one. (2) $\mu$ can-not be decomposed as ...
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2answers
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What is $T^1(\mathbb H^2/PSL_2(\mathbb Z))$?

Let $\mathbb H^2$ be the upper-half plane. The group $PSL_2(Z)$ acts on $\mathbb H^2$ by isometries, and hence we get an action on $T^1(\mathbb H^2)$. This action is free, smooth, and proper, and thus ...
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Question about proof of equivalent conditions for ergodicity

I am currently attempting to work through Walters' text on Ergodic theory and in the proof of the list of equivalent conditions for ergodicity they state $T^{-n}B\triangle B\subset\cup_{i=0}^{n-1}T^{...
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Is the image of a metric automorphism under a measure-preserving mapping also a metric automorphism?

I'm having my first tiny little bits of ergodic theory so please forgive the probable naivete of the question. So, I'm looking at the first page of chapter 8 of Cornfeld, Fomin, Sinai's "Ergodic ...
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Measure Theory: Simple Measure Invariance Proof

Let M be a metric space, $f: M \rightarrow M$ be a measurable transformation and $\mu$ be a measure on M. Show that $f$ preserves $\mu$ if and only if $\int \phi d\mu =\int \phi \circ f d\mu$ for ...
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Number of preimage in a ergodic map

The problem is to show that if I have $T: (X, \mu) \rightarrow (X, \mu)$ that preserves $\mu$ and is ergodic then the number of preimages of a point is well-defined and is constant $\mu$-ae. I ...
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Definition of ergodic map

I ask the similar question before. About definition of Ergodic theorem. Now just sincerely ask another fundamental problem about the definition of ergodic map. The following definition is what I ...
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$\mathbb{Z}^2$-action on product is ergodic

I try to solve exercise 8.1.1 from Einsielder's book Ergodic Theory. It is: "Let $(X,B_X, μ, T)$ and $(Y,B_Y , ν, S)$ be ergodic Z-actions. Define a $\mathbb{Z}^2$-action on the product $(X×Y, μ×ν)$ ...
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Proving that the conditional entropy of a probability measure is concave

Let $\mu$ be a probability measure on $\mathcal{X}$ and let $\mathcal{E}, \mathcal{F}$ be countable partitions of the space. Define the entropy of $\mu$ with respect to the partition $\mathcal{E}$ as $...
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Dynamical system with $f(x)\leq f(Tx) \leq f(T^2 x) \leq …$

Let $(X,\mathcal{F},\mu,T)$ be a measure preserving system and $f\in L^1(X,\mathcal{F},\mu)$. Problem: show that if $f(x)\leq f(Tx)$ $\forall x\in X$ then $f(x)=f(Tx)$ $\mu$-a.e. It seems to be ...
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If $\mu\neq \nu$ then $\frac{1}{2}(\mu+\nu)$ is not ergodic

The following problem looks simple, but it seems there is more needed than just using the definition of ergodicity. Let $(X, \mathcal{F}, \mu, T)$ and $(X, \mathcal{F}, \nu, T)$ be two measure ...
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If $T\times T$ is ergodic, then so is $T\times T\times T$

Given a measure preserving system $(X, \mathcal{F}, \mu, T)$, show that $T \times T$ is ergodic with respect to $\mu\times \mu$ if and only if $T \times T \times T$ is ergodic with respect to $\mu\...
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Spectral Theorem for Unitary Operator

It is well known that the following - in many literature - called the Spectral Theorem for Unitary Operator. I would like to know where i can find further information about it and its proof.
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43 views

Show that i.i.d. process is ergodic

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(E,\mathcal E)$ be a measurable space $(X_n)_{n\in\mathbb N_0}$ be an $(E,\mathcal E)$-valued i.i.d. process on $(\Omega,\mathcal A,\...
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Construction of a (jointly?) stationary and ergodic vector sequence.

In a recent effort to understand stationary ergodic processes, I stumbled upon a paper that leaves me somewhat puzzled. I would be very grateful for any pointers. The line of reasoning is as follows: ...
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1answer
42 views

Khintchine's Recurrence Theorem

Yves Coudène. Ergodic Theory and Dynamical Systems. Page 14. Roughly speaking he mentioned that the Khintchine's Recurrence Theorem is a generalisation of Poincaré's Recurrence Theorem. How can ...
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A sufficient condition for minimality in a Topological Dynamic System

Am trying to prove the following : Let ($X,T$) a uniquely ergodic topological system.We suppose that the uniquely $T-$invariant measure $\mu$ satisfies the property :$\mu(U)>0$ for every non-...
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1answer
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Is this map chain-transitive

Let $f:\mathbb R^2→\mathbb R^2$ be a continuous self-map and let $δ$ be a positive real number. A (finite or infinite) sequence $(x_{n})_{n≥0}$ is a $δ$-chain if $$d(f(x_{n}),x_{n+1})<δ$$ for all $...
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An entropy inequality: $h_{\mu}(\beta,T)\leq h_{\mu}(\alpha,T)+H_{\mu}(\beta|\alpha)$

Let T be a measure preserving transformation on the probability space $(X,\mathcal{F},\mu)$. I have already solved this problem: Suppose $\alpha$ is a finite partition of $X$. Show that $h_{\mu}(\...
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How to show that q-coloring graph is ergodic

Informal: I want to show that a q-coloring of graph $G$ is ergodic (i.e. strongly connected and aperiodic) Formally: For a given graph $G(V,E)$ where $|V|=n$ with maximum degree $\Delta\geq1$. Also, ...
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Birkhoff average of $x \mapsto x+1$ in $\mathbb R$ with $L^p$ observable

Let $f \in \mathcal L^p(\mathbb R, \lambda)$, where $\lambda$ is Lebesgue measure and $p \in (1,\infty)$. And let $T : \mathbb R \to \mathbb R$ be the map $T(x) = x+1$. I want to show: $$ \frac 1 n \...
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Understanding compact extensions and almost-periodic functions

This question comes from my attempt to understand theorem $7.21$ in E-W. This concerns the dichotomy between relatively weak-mixing extensions and compact extensions. I cannot understand the proof as ...
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The Ergodicity of Baker's transformation

How to prove that Baker's transformation is ergodic with respect to Lebesgue measure directly without showing that it is isomorphic to bernoulli shift ?
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About function V in geometric drift condition for Markov Chain

When I read the geometric ergodicity of Markov chain in Meyn and Tweedie, I note that in the drift condition $PV(x)<\lambda V(x)+bI_{x\in C}$, where $V(x)\ge 1$ is required. Why $V(x)\ge 1$ rather ...
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31 views

Equidistribution of $\{p_n^2 \alpha \}$

Let $p_n$ be the $n$th prime and $\alpha$ an irrational number. Vinogradov proved that the sequence $\{p_n \alpha \}$ is equidistributed. Is it known whether the sequence $\{p_n^2 \alpha \}$ is ...
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How to deal with sets of measure zero?

When I learned measure theory, sets of measure zero begin to perplex me somewhat, especially the Fubini Theorem. My method to deal with them is to consider equivalent classes of measurable functions ...
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Does a primitive transition matrix implies that the chain is ergodic?

Let matrix $A$ be a primitive matrix so $A^k>0$ and $A$ is also a transition (stochastic) matrix . Can we say that $A$ is ergodic?In other words, can we say that $A$ is: (1) strongly connected (...
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Recurrent Markov chain has an invariant measure

$\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\mc}{\mathcal}$ $\newcommand{\P}{\mathbb P}$ $\newcommand{\N}{\mathbb N}$ $\newcommand{\E}{\mathbb E}$ $\newcommand{\mr}{\mathscr}$ I am looking for the ...
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1answer
45 views

Can ergodicity be checked by restricting attention to continuous maps?

The following is well-known: Theorem. Let $(X, \mathcal F, \mu)$ be a probability space and $T:X\to X$ be a measure preserving map. Then the following are equivalent: $\bullet$ $T$ is $\mu$-...
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The action of $SO(n + 1)$ on $S^n$ is ergodic

I want to prove that this action is ergodic. That is, if $\lambda$ is the normalized Lebesgue measure on $S^n$ and $A$ is a measurable set such that for all $g \in SO(n + 1)$ we have $\lambda(gA \...
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What is the mixing time of a random walk of a rook

Let $G(V,E)$ be the following graph: The vertex set $V$ is a $n\times n$ grid, and two vertices are connected $(E)$ if they lie on either the same row or the same column. This is the rook's graph: It ...
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Markov measure on cylinders

If T is a Markov map with associated partition $\{A_i\}$ and $A_{i_0i_1i_2...i_{n-1}}$ an n-cylinder, i.e. $A_{i_0i_1i_2...i_{n-1}}:=A_{i_0}\cap T^{-1}A_{i_1}\cap T^{-2}A_{i_2}\cap ...\cap T^{-(n-1)}...
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How to prove that the Modified Boole Transformation preserves the following measure?

Operator Theoretic Aspects of Ergodic Theory. T. Eisner. B. Farkas. M. Haase and R. Nagel. Page 90 Exercise 3. How to prove that the following transformation $$T:\mathbb{R}\rightarrow \mathbb{R}$$ $$T(...
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Examples of non-abelian simply connected nilpotent Lie groups.

I am searching for examples of connected locally compact group $G = N \rtimes H$, where $N$ is a simply connected nilpotent non-abelian Lie group, $H$ is linear reductive and $H$ operates on $N$ ...
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1answer
77 views

Countable open cover up to a null set

Given a metric space $(X,d)$,a probability measure $\mu$ (on the Borel sigma algebra) and an open cover $C:=\{A_i\}_{i\in I}$ of $X$, is it always possible to find a countable subset of $C$ that ...
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How to interpret a theorem stating that orbits are “uniform on average”

I am reading through these notes: http://wwwf.imperial.ac.uk/~dcheragh/Teaching/2015-F-DS-MPE.pdf There is a theorem (Theorem 1 under the section Distribution of Orbits) saying the following: Here $...
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What is the difference between ergodicity and the law of large numbers?

I want to begin by saying that I know absolutely no measure theory. To my knowledge, roughly speaking a stochastic process is ergodic if its time average converges to the expectation (space average) ...
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Topological conjugacy between dyadic map and tent map

For trying to prove that the tent map $$T(x)= \begin{cases} 2x &\text{ if } x\in[0,\frac{1}{2}]\\ 2-2x &\text{ if } x\in[\frac{1}{2},1] \end{cases} $$ is ergodic, I have already shown that ...
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1answer
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Strong Folner condition(SFC) implies the existence of a left Følner sequence.

I got stuck with this problem while reading Density in Arbitrary Semigroups by Hindman and Strauss. It says: Problem: If $S$ is a countable semigroup. Then SFC on $S$ implies the existence of a left ...