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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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What does mod 0 mean in the context of ergodic theory?

I have come across the following definition: A measure-preserving transformation (or flow) $T$ is ergodic if any essentially $T$-invariant measurable set has either measure 0 or full measure. ...
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36 views

Approximating a real arbitrarily well by $Z$-linear combination of two reals having irrational ratio.

$\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\Z}{\mathbb Z}$ $\newcommand{\R}{\mathbf R}$ Let $\alpha$ and $\beta$ be positive real numbers such that $\alpha/\beta$ is irrational. Then the following ...
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1answer
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In a Banach space $x_n\text{cos}(nt)+ y_n\text{sin}{nt}\rightarrow 0$ implies both sequences individually go to $0$.

Let $X$ be a Banach space, and let $\{x_n\}$, $\{y_n\}$ be two sequences in $X$. Suppose $x_n\text{cos}(nt)+ y_n\text{sin}(nt)\rightarrow 0$ as $n \rightarrow \infty$ for all $t$ in a non degenerate ...
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Kolmogorov-Sinai theorem for a generator with infinite entropy

I can find many books and documents stating the Kolmogorov-Sinai theorem (that is $h(T) = h(T,P)$ if $P$ is a generating partition) when the generating partition is finite or countable with finite ...
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14 views

Proving Conservativeness directly from Maharam's theorem

From the book Infinite ergodic theory of numbers, Kesseböhmer, proposition 2.4.27b on page 106. I'm sorry but I don't have a link. They say it follows directly from Maharam's theorem. They probably ...
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46 views

Compact, infinite, invariant set on shift space.

Let $X \subset \{0,1\}^{\mathbb{N}}$ be a compact, infinite, shift invariant set. Does $X$ contains a non-periodic point?. My attempt: I was trying to construct a non-periodic point, given that we ...
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42 views

“Two step” Markov chain is actually a Markov chian

Let $X$ be a compact metric space and $\mathcal X$ be its Borel $\sigma$-algebra. Let $\mathscr P(X)$ be the set of all the Borel probability measures on $X$. A Markov chain on $X$ is a measurable map ...
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Equivalence/Relationship between the two Krylov-Bogolyubov theorems.

When looking at the Wikipedia page and dynamical systems literature it seems that there are two theorems called the 'Krylov Bogolyubov' theorem. One for Markov processes and another in Ergodic theory. ...
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1answer
33 views

Measurability with respect to the $\sigma$-algebra of invariant sets.

Let $(X, \mathcal F, \mu)$ be a measure space and $T:X\to X$ be a measure preserving transformation. Let $$\mathcal E=\{E\in \mathcal F:\ \mu(T^{-1}(E)\ \Delta\ E) = 0\}$$ where $\Delta$ denotes the ...
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Trouble understanding the proof of disintegration of measure by Tao

Theorem $4$ of this blog entry of Terrence Tao states the following: Let $X$ be a compact metric space, $\mathcal X$ be the Borel $\sigma$-algebra of $X$, and $\mu$ be a probability measure on $X$. ...
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1answer
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Entropy Theory and Conditional Entropy

Peter Walters An Introduction to Ergodic Theory. Chapter 4 Entropy Page 83 & 84 How did they duduce the last formula ?
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Counterexample for the $L^p$ Ergodic Theorem of Von Neumann when $p=\infty$

I am looking for a counterexample to the following theorem when $p=\infty$: $L^p$ Ergodic Theorem of Von Neumann. Let $1\leq p<\infty$ and let $T$ be a measure-preserving transformation of ...
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2answers
70 views

Example of measures being ergodic but not invariant

Let $\mu$ be a measure, $f:X\rightarrow X$ being a map and $\Sigma$ is a $\sigma$-algebra. $\mu$ is $f$-invariant if $\mu(E) = \mu(f^{-1}(E)), \forall E\in \Sigma$. $\mu$ is ergodic with ...
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1answer
24 views

About definition of Ergodic theorem

Let $(X,\Sigma, \mu)$ be a probability space, and $T:X\rightarrow X$ be a measure-preserving transformation. We say $\mu$ is ergodic with respect to $T$ if for every $E\in \Sigma $ with $T^{-1}(...
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1answer
129 views

On the definition of ergodicity and how it relates to random processes.

Let $\mathcal{M} : = \{ \mu | \mu \text{ is a probability measure} \}$, and let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a measurable function. $\mu \in \mathcal{M}$ is ergodic if for all $A \in \...
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If $T:[0,1] \rightarrow [0,1]$ preserves Lebesgue, then $\liminf_n(n|T^n(x)-x|) \leq 1$ [closed]

Let $T:[0,1] \rightarrow [0,1]$ be a measurable function such that $T$ preserves Lebesgue, then for almost all point: $$\liminf_n(n|T^n(x)-x|) \leq 1$$
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1answer
37 views

A strengthened implication of the ergodicity

Let $( X , Σ , μ )$ be a probability space, and $T : X → X$ be a measure-preserving transformation. We say that $T$ is ergodic with respect to $μ$ if for every $E ∈ Σ$ with $T^{-1}(E)=E$ either $μ ( ...
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1answer
33 views

Spectral Invariants

Peter Walters An Introduction to Ergodic Theory Chapter 2 Page 66 What does the idea (Property P) mean ? I couldn't understand it
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The Generalisations of Ergodic Theorems

We all know that the first two ergodic theorems are Birkhoff ergodic theorem and Von-Neumann ergodic theorem. And we also have Wiener-Wintner ergodic theorem. Are these theorems all the ergodic ...
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Confusion on construction of type III factors

Well $G$ is group of translations: $\mathbb{R} \ni s:\mapsto as+b $ where $a, b \in \mathbb{Q}$, modulus of $a\neq1$. Under this action of $G$ on $\mathbb{R}$ why Lebesgue measure is not invariant? ...
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Is “shadowing” invariant under topological conjugacy?

A homeomorphism $f: X \rightarrow X$ of a metric space is said to have the shadowing property if for all $\varepsilon > 0$ there is a $\delta>0$ such that for every sequence $(x_n)_{-\infty}^{\...
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prove ergodic theorem for finite irreducible, aperiodic Markov Chain

State and prove ergodic theorem for finite irreducible, aperiodic Markov Chain with transition probability matrix $P=(p_{ij})$. I know what irreducible, aperiodic means. But I do not understand about ...
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1answer
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Bernoulli measure is aperiodic

Let $\Sigma=\{s_1,\dots,s_m\}$ be a finite list of symbols, and put $X=\Sigma^\mathbb{Z}$. Consider the left two-sided shift $T:X\to X$ given by $T(x_n)=(x_{n+1})$. Given an $m$-dimensional vector $\...
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1answer
84 views

Proof of Birkhoff ergodic theorem

The proof of Birkhoff ergodic theorem in the book of Peter Walters; An introduction to Ergodic Theory. Page 39. The second case when $m(X)=+\infty$. After the sentence (The function $H_N$ ...) I ...
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Products of Positive Matrices

Is it true that every infinite product of positive matrices (every element is larger than 0) converges to a rank one matrix? I learned that Birkhoff's Contraction Coefficient is smaller than one for ...
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Ergodicity bounds for Matrix Products

I have a backward product of $n \times n$ matrices. Every row except the first and the last row are stochastic. The only entries which differ between the matrices are $x_i$ and $y_i$. For every ...
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1answer
98 views

Birkhoff sum converges

I am master student.I have been starting to learn Ergodic theory. Let $(X,A,\mu,T)$ be a measure-preserving dynamical system. Birkhoff Ergodic theorem $$\lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1}...
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1answer
20 views

Small question in a proposition involving Ergodicity

Ergodicity -- A measure preserving transformation $T$ on the space $(X, \mathcal{B} , \mu)$ is called ergodic iff $\forall B \in\mathcal{B}$ satisfying $T^{-1} B = B$ we have $\mu(B) = 0$ or 1. ...
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What does $A\cap T^{-n}(B)$ imply about $T^n(A)\cap B$?

From https://www.staff.science.uu.nl/~kraai101/LectureNotesMM-2.pdf example 1.8.4 (multiplication by 2 modulo 1), page 29. What do they do at the second equality $\lambda(A\cap T^{-n} B) = \frac{1}{\...
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1answer
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If $T_A$ is ergodic and $\mu\Big(\cup_{n\in\mathbb{N}} T^{-n} A\Big)=1$ then $T$ is also ergodic.

This is exercise 1.8.4 from https://www.staff.science.uu.nl/~kraai101/LectureNotesMM-2.pdf "Show that if $T_A$ is ergodic and $\mu\Big( \cup_{k\geq 1} T^{-k}(A)\Big)=1$ , then, $T$ is ergodic."
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1answer
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Unitary representation of amenable groups

Suppose $\Gamma$ is an amenable group, let $\pi:\Gamma\mapsto \mathcal{u}(\mathcal{H})$ and $\rho:\Gamma\mapsto \mathcal{u}(\mathcal{K})$ are two unitary representations of $\Gamma$, what does one ...
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1answer
30 views

Are all invariant measures to a Markov Process found by following the process?

Some (hand-wavy) context: Suppose that we have found the invariant measures to a Markov Process, $(X_t)_{t=1}^\infty$, on a compact set $S$ by the following the process itself, i.e. looking at the ...
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Geodesic flow on Riemannian manifolds

A naive question. I see many research papers on geodesic flows on Riemannian manifolds. Some of them try for instance to show that they are ergodic. It is probably a very broad question but could you ...
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Entropy of a Measure Preserving Transformation

$\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\lrp}[1]{\left(#1\right)}$ I am reading the concept of entropy from Peter Walters An Introduction to Ergodic Theory and I am having trouble understanding ...
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1answer
57 views

ergodicity of irrational rotation of product

I need some help with the following. Suppose $(X,\mathcal{B},\mu,R)$ is an ergodic measure preserving dynamical system. Consider the torus $Y=\mathbb{R}/\mathbb{Z}\times\mathbb{R}/\mathbb{Z}$ and ...
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Density of the sequence $\sqrt2^{\,n}$ modulo 1

I want to know whether the fractional parts of the sequence $i\mapsto \sqrt{2}^i$ are dense in $[0, 1)$. Obviously this is the same as the sequence $i\mapsto \sqrt{2}\cdot2^i$. I did the obvious thing ...
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ergodic system has dense orbits a.e., continuity of T

My question is related to this question, i.e. under which conditions we have that $\mu$-a.e. point has a dense orbit in a dynamical system $(X,T)$ with measure $\mu$. I am confused about the ...
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1answer
53 views

Where does this proof of ergodicity fail?

I am a bit confused and was hoping the folk of MSE would help (and point out where the error is made). Suppose $(X,\mathcal{B},\mu,R)$ and $(Y,\mathcal{C},\nu,S)$ are ergodic measure preserving ...
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1answer
77 views

If $T$ has continuous spectrum and $(f, 1)=0$ then $\mu_{f}$ has no atoms.

I'm reading Peter Walters' An Introduction to Ergodic Theory, and I don't understand his remark. Following is a sort of lemma, called Spectral Theorem for Unitary Operators, to prove the equivalence ...
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1answer
198 views

Your Favourite Application of the Birkhoff Ergodic Theorem

Here we have a big list of great applications of the Baire category theorem. I recently read the Birkhoff ergodic theorem and I think perhaps this theorem is on par with Baire's theorem in terms of ...
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1answer
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How important is ergodic theory in fundamentally understanding statistical mechanics?

Recently, I realized the physics course in my uni has been lacking mathematical rigor and I have been attempting to compensate by adjoining it with personal mathematical study. For example, learning ...
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1answer
73 views

$\mu $ ergodic $\iff \mathcal{O}^{+}_{\phi}(x)\cap A\neq\emptyset$

Let $\mathcal{A}$ be a $\sigma$-Algebra on $X$ and $\phi: X\rightarrow X$ measurable (i.e. $\phi^{-1}(A)\in\mathcal{A}$) . Let $\mu:\mathcal{A}\rightarrow[0,1]$ be probability measure which is $\phi$-...
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1answer
58 views

Is this formula for entropy true?

Let $T:X\to X$ a ergodic transformation in a measure space $(X,\mu)$ and let $A\subseteq X$ with positive measure. If $\mathcal{A}=\{\alpha\subseteq 2^X\mid\alpha\text{ is a finite partition with }X\...
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Equivalence between non-singular vector field without periodic orbits and the irrational flow on a torus.

I'm reading the book "The dynamics of vector Fields in dimension 3 - Matthias Moreno and Siddhartha Bhattacharya " On page 6 the authors enunciate the following theorem: Theorem:(Poincaré, Denjoy:)...
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1answer
33 views

Ergodicity of surjective continuous endomorphism of compact abelian group

Ergodic Theory with a view towards Number Theory Manfred Einsiedler and Thomas Ward Page 31 Why if $f$ is invariant then we have the following equalities in the box
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Ergodicity properties of diffusions: equivalent properties and criterias.

I have just started reading about the Ergodicity of Markov processes, and I'm particularly interested in showing that time averages of certain diffusions converge as time grows, and in characterizing ...
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41 views

Ergodic Markov chains and eigenvalues

I just read on wikipedia that a way to check whether a Markov chain is ergodic is to compute the eigenvalues of the transition matrix, and if those are all (except for 1) less than 1, then the chain ...
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1answer
30 views

Characterization of tail $\sigma$-algebra and symmetries of $\{0,1\}^{\mathbb{N}}$

Recently the following question was asked: If a measurable subset of $\{0,1\}^{\mathbb{N}}$ (with the Borel $\sigma$-algebra) is invariant under the action of $\mathbb{Z}_{2}^{\oplus \mathbb{N}}...
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1answer
54 views

Topological transitivity+contraction

I'm looking for a hint: Suppose $T:X→X$ is a dynamical system. Assume that $T$ is both topologically transitive and a contraction, i.e. $d(T(x),T(y))≤d(x,y)$ for all $x,y∈X$. Prove that T is minimal. ...
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1answer
54 views

Dynamical system question

I'm looking for a hint on this question: Suppose $X$ is a metric space with at least one isolated point and $T:X\to X$ is a topologically transitive dynamical system. Show that $X$ is necessarily ...