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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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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
54 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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1answer
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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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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 ...
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35 views

Equivalence of definitions of ergodic action

Let $G$ be a group acting on a probability measure space $(X, \mu)$ by measure-preserving transformations. I have read the two following definitions of ergodicity of such an action: For every ...
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Ergodic action of dense subgroup

Let $G$ be a group acting ercodically on a probability measure space $(X, \mu)$. Let $\Gamma$ be a countable dense subgroup of $G$. Is the action of $\Gamma$ also ergodic? The case I am interested in ...
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Which of the open problems in Halmos's Lectures on Ergodic Theoy are still open today?

For a while now, my bedtime reading has been a recent Dover reprint of Paul Halmos's 1956 booklet Lectures on Ergodic Theory. It's an interesting little work. At the end he lists a number of open ...
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Dynamic system $f(x) = 2x$ mod $1$

I am reading the following paper: ergodic theory of chaos and strange attractors, by J.-P. Eckmann (can be easily downloaded) My question is from an example on p....
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46 views

Are factors corresponding to a sub-$\sigma$-algebra unique?

Let $X$ be a compact metric space and $\mathcal B$ be its Borel $\sigma$-algebra. Let $\mu$ be a Borel probability measure on $X$ and $T:X\to X$ me an invertible measure preserving transformation. Let ...
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1answer
107 views

Question regarding motivation of spectral theorem for unitary operators

$\newcommand{\mc}{\mathcal}$ $\newcommand{\ab}[1]{\langle #1\rangle}$ Theorem B.4 in Einsiedler and Ward's [EW] Ergodic Theory with a view towards Number Theory states the following: Theorem 1. ...
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48 views

Continuity(?) of induced group representation in the isometries of $L^p$

Let $G$ be a topological group which acts on a measure space $(X,\mu)$ by measure preserving transformations. It's well known that this induces a representation $\pi$ of $G$ in $O(L^p (X,\mu))$, the ...
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Clarification on a proof of Roth's theorem

Roth's theorem is stated in the book by Einsiedler and Ward, theorem 7.14 page 191 as: Let $(X,\mathcal{B},\mu,T)$ be a measure-preserving probability system. Then, for any functions $f_1,f_2 \in L^{\...
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Can this vague feeling about entropy be made precise?

Let $I$ denote the unit interval and $\mu$ be the Lebesgue measure. Let $S:I\to I$ be the map defined as $S(x)=2x \pmod{1}$. Then it is known that for any measurable subset $A$ of $I$ we have $$ \lim_{...
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Proving ergodicity of map on circle

This dynamical system: $x_t=x_0 + \omega t $ mod 1 Should be periodic if $\omega$ is rational and ergodic if irrational. I know that if the time average of generic smooth function $A(x)$ ...
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Anzai flow in noncommutative geometry

Consider Anzai flows (cf. Anzai: Ergodίc Skew Product Transformations on the Torus, Osaka Math. J. 3 (1951), 83-99) on the two dimensional torus $T^2$. I would like to know if there exists some ...
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How does proximality relates to p- limit?

Please help me to solve a problem given in the survey Minimal Idempotents and Ergodic Ramsey theory by Vitaly Bergelson(Exercise 15(iii), page 23), which is Problem: Prove that if $x_1,x_2$ are ...
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A certain fact about conditional expectation concerning the one sided shift associated to the random walk on a graph.

$\newcommand{\mc}{\mathcal}$ $\newcommand{\E}{\mathbf E}$ $\newcommand{\R}{\mathbf R}$ $\newcommand{\P}{\mathbb P}$ $\newcommand{\wh}{\widehat}$ Definitions Let $G$ be a connected graph on a finite ...
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Sufficient condition for square root fluctuations of an ergodic sequence

Suppose I have a random sequence $\mathbf{X}=\{X_n\}_{n\in\mathbb{Z}}\subset \mathbb{R}^{\mathbb{Z}}$ that is ergodic with respect to translations. I am interested in a sufficient condition on $\...
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Is $(f,\mu)$ weakly mixing?

I know how to show that if $(f,\mu)$ is weakly mixing then so is $(f^{k},\mu), \forall k \geq 2$. I was trying to proof the converse. I don't even know if this result is true and, at the end, seems to ...
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How to calculate the envelope of the trajectory of a double pendulum?

Consider a double pendulum: Background For the angles $\varphi_i$ and the momenta $p_i$ we have (with equal lengths $l=1$, masses $m=1$ and gravitational constant $g=1$): $\dot{\varphi_1} = 6\frac{...
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Why does ergodicity fail?

Apparently the following is not ergodic, and a very sketchy argument is given, so I was hoping someone here would be able to explain/give a better argument. Suppose $G$ is a compact connected non-...
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1answer
87 views

Almost all non-negative real numbers have only finitely many multiple lies in a measurable set with finite measure

Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that Lebesgue measure of $A$ is positive i.e. $0<\lambda(A)<\infty$. Let $S$ be the set defined as follows: $$S:=\{t\in [0,\infty):nt\...
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1answer
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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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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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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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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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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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“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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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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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
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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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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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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
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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
58 views

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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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 ...