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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Connectedness in Measure Preserving Ergodic Systems

In topology a space $X$ is called connected if there is no partition of $X=A\cup B$, that $A$ and $B$ are open. Since ergodic systems have the property that each trajectory visits any neighbourhood of ...
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100 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
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32 views

rising sun inequality [duplicate]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
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1answer
43 views

Question about Kronecker factor

In his paper Ergodic methods in additive combinatorics, Bryna Kra said that the Kronecker factor $(Z_1, \mathcal{Z}_1, m, T)$ of $(X, \mathcal{X},\mu,T)$ is the sub-$\sigma$-algebra of $X$ spanned by ...
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2answers
79 views

Does any measure preserving system have an invertible extension?

Let $\mathsf{X} = \left\{ X,\mathcal{B},\mu,T \right\}$ be any measure preserving system. A sub-$\sigma$-algebra $\mathcal{A}\subseteq \mathcal{B}_X$ with $T^{-1}\mathcal{A}=\mathcal{A}$ modulo $\mu$ ...
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1answer
23 views

How do you get a measure space out of a dynamical system?

I'm reading a book on ergodic theory (by Cesar Silva), and also have read Stein and Shakarchi's third book on undergraduate analysis, where there is a section devoted to some ergodic theory. Both ...
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3answers
100 views

Showing $\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \log(\cos(\phi))\cos(\phi) \ d\phi = \log(4) - 2 $

This is a minor detail of a proof in 'Chaotic Billiards' by Chernov and Markarian which I foolishly decided to verify. It's page 44 of the book, during the proof that lyapunov exponents exist almost ...
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1answer
68 views

Ergodic Rotation of the Torus

Consider the measure preserving dynamical system $(\mathbb{R}^2 / \mathbb{Z}^2, \mathcal{B} \otimes \mathcal{B}, \lambda \otimes \lambda, R_{(\alpha, \beta)})$. This is the torus with the borel ...
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4answers
73 views

Rigour and Formal Reference for Ergodic Theory

I am not even a beginner to Ergodic Theory, but I want to start to read about it. I am coming from a math background and for me its quite important that the definitions to be stated and the formalism ...
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1answer
30 views

Mean Ergodic Theorem for ergodic transformations

I understand how the ergodic averages for an $L^2$ function converge in norm to the orthogonal projection on the space of invariant functions, and I understand how for ergodic transformations this ...
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0answers
36 views

Why define shift invariant set as $\Lambda=\tau^{-1}\Lambda$?

Can we define shift invariant set as $\Lambda = \tau\Lambda$ instead of $\Lambda = \tau^{-1}\Lambda$, where $\tau$ is the shift operator? Can permutable set be defined as either $\Lambda = \pi ...
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30 views

A variation of Birkhoff's Ergodic theorem

Suppose $T_k$ are measure perserving for each $k$, and $f$ is integrable. Is it true that $\frac{1}{n}\sum_{k=1}^n f(T_k\circ T_{k-1}\dots\circ T_1 \omega)$ converge almost surely?
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1answer
41 views

What is “abstract” ergodic theory?

This is just a question about the usage of the term "abstract". What kind of questions in ergodic theory is considered "abstract" and what's a "regular" question? From some seminars it seems that ...
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2answers
55 views

Measure preserving transform and convergent random variables

I have been trying to learn about Ergodic Theorem for a while and now I have a problem I can't solve. Assume $T$ is a measure preserving transform and $X_n\rightarrow X$ everywhere. Also, assume ...
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0answers
16 views

$L_1$ mean ergodic theorem for the action of compact group

Let $X$ be a Polish space with Borel probability measure $\mu$. A compact group $G$ acts on X continuously. It is right that for any $f\in C_b(X)$ exists a sequence $(g_k \in G)_{k\in \mathbb{N}}$ ...
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1answer
51 views

Derivation of the Mexican wave as Mean field equilibrium of an ergodic control problem

I'm dealing with a Mean Field system introduced by Gueant, Lasry, Lions in their "Mean Field Games and Applications" which admits as solution the so-called Mexican wave. Without going into the details ...
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1answer
60 views

What keeps measure-preserving transformations from concentrating in a particular portion of a probability space?

I'm trying to show that for an event A with positive probability there is some n bounded by 1/P(A) such that $P(A \cap$ T$^{-n}A) > 0$, where T is a probability-preserving transformation. I'm ...
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0answers
39 views

What is the density of the SRB measure conditioned to unstable manifolds?

I have a question regarding the SRB measure. As Lai-Sang Young puts it, the SRB-measure is the invariant ergodic invariant measure most compatible with volume. (see [ ...
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0answers
40 views

Cesàro Sum of Tangent

Can you proof or disprove the following? $\lim_{n \to \infty} (\frac {\tan1+\tan 2+\cdots+\tan n}{n})=0$. Is there any ergodic type theorem that can come to help?
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1answer
32 views

Question concerning the proof of the Ergodentheorem by Birkhoff

Let $(\Omega,\mathcal{A},\mu,T)$ be an ergodic dynamical system and $f\in L_{\mu}^1$. Then it is a.s. $$ \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f\circ T^k=\int f\, d\mu. $$ Now I ...
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0answers
62 views

Circle rotation (dynamic system)

Here's a passage of my script I do not understand. Define $\Omega:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\}$ and consider $a\in [0,1)$. Then $$ T_a:=\Omega\to\Omega, z\mapsto ...
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1answer
31 views

ergodic theorem in the seasonal component analysis of time series

When studying the "seasonal components" part of time series, I once read the following statement. I do not understand what role does the ergodic theorem play here? The decomposition of the process ...
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0answers
24 views

Unique ergodicity a spectral invariant?

Suppose $f:X\to X$ and $g:Y\to Y$ are two automorphism of measure spaces. Suppose there is an unitary operator $V:L^2(X)\to L^2(Y)$ satisfying $VU_fV^*=U_g$, here $U_f$ and $U_g$ are the Koopman ...
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83 views

Measurability of points regular

I'm reviewing the proof of the theorem of oseledet the book Mañe: Let $M$ a compact metric space and $f:M \rightarrow M$ a homeomorphism, $\pi: F \rightarrow M$ a finite-dimensional continuos vector ...
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1answer
88 views

Is there a characterization of the shift-invariant ergodic measures?

Consider probability measures $\mu$ on the space $\{0,1\}^\mathbb{N}$ that are shift-invariant with respect to the left-shift map. Is there a nice characterization of the ergodic shift-invariant ...
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0answers
29 views

Krylov-Bogoliubov theorem without continuity

This question is very closely related to: Continuity in the Krylov-Bogoliubov theorem. The standard counterexample, which is presented in Katok-Hasselblatt is the following: Let $f:[0,1]\rightarrow ...
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0answers
39 views

Birkhoff averages

Let $(X, \mathcal{A},\mu)$ be a probability space and $T:X\rightarrow X$ an ergodic transformation. The Birkhoff averages of a function $\phi:X \rightarrow \mathbb{R}$ are defined by $$ ...
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1answer
43 views

E. Artin theorem? (Ergodic theory)

In the framework of mathematical cosmology, Bianchi IX model has great importance due to its stochastic properties. I'm reading a publication in which is claimed The use of the invariant measure ...
4
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1answer
76 views

Ergodic Process: Does it visit all state?

I read in this article: " Ludwig Boltzmann, coined "ergodic" as the name for a stronger but related property: starting from a random point in state space, orbits will typically pass through every ...
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1answer
44 views

Koopman is not surjective

I want to prove that the Koopman operator $U_f : L^2 (\mu) \rightarrow L^2 (\mu)$ such that $U_f(\phi) =\phi \circ f$ is not surjective. Where $ \mu $ is a measure preserving mapping $f$. I was ...
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1answer
43 views

Pointwise Ergodic Theorem - one particular estimate

I am struggling with an estimate in a proof of the pointwise ergodic theorem discussed in T. Ward and M. Einsiedler: Ergodic Theory with a view towards Number Theory, which is left as an exercise. I ...
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1answer
23 views

Show by induction, that $\mu(T^{k}A\Delta A)=0~\forall~k\in\mathbb{Z}$

Here are some definitions that might be necessary for my following question: (I) The Quadruple $(\Omega,\mathcal{A},\mu,T)$ is called dynamical system, if $(\Omega,\mathcal{A},\mu)$ is a probability ...
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1answer
52 views

How can I prove this equivalence concerning ergodicity?

I write you, because I have a problem to show two equivalences. But before writing them down, I give you all the definitions we had as background: (I) The Quadruple $(\Omega,\mathcal{A},\mu,T)$ is ...
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1answer
45 views

Ergodic: Show another implication

Let $(\Omega,\mathfrak{A},\mu,T)$ be a dynamic system in measure theory and $p\geq 1$. Show that the following statements are equivalent: $$ (1)~~~~~(\Omega,\mathfrak{A},\mu,T)\text{ is ...
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1answer
40 views

Ergodic system, show an implication

Let $(\Omega,\mathfrak{A},\mu,T)$ be a dynamic system in measure theory and $p\geq 1$. Show the implication $(1)\implies (2)$, whereat $$ (1)~~~~~\forall f\in L_{\mu}^p: f= f\circ ...
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1answer
35 views

Show: Ergodic implies $f=\text{const a.s.}$

Let $(\Omega,\mathcal{A},\mu,T)$ be a dynamic system in measure theory. Let this system be ergodic. Show then then this implies $$ \forall f\colon\Omega\to\mathbb{R}\mbox{measurable}: ...
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1answer
74 views

Ergodic dynamic system (show equivalences)

Let $(\Omega,\mathcal{A},\mu,T)$ be a dynamic system in meaasure theory. Show that the following statements are equivalent: (1) $(\Omega,\mathcal{A},\mu,T)$ is ergodic (2) $\forall A\in\mathcal{A}: ...
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1answer
23 views

Testing density with a countable family

Let $d$ denote the lower density on $\mathbb{N}$, $a>0, $ $\mathbb{N}_{a}:=\left\{ B\subset\mathbb{N}:{d}(B)\geq1-a\right\} $ and $A\subset\mathbb{N}$. If $A\cap B\neq\emptyset$ for every ...
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0answers
37 views

G-space decompositions preserved by equivariant maps?

Let $X,Y$ be topological $G$-spaces, with (left) $G$-invariant probability measures $\mu_X,\mu_Y$ respectively, and let $f:X \to Y$ be a surjective $G$-equivariant map preserving the measures, i.e. ...
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2answers
75 views

Examples of ergodic geodesic flow

Are there any good examples of a geodesic flow that is ergodic? I know the result that states that the geodesic flow for manifolds with negative curvature are ergodic, but I'm fishing for some ...
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0answers
80 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
2
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1answer
25 views

almost periodic visits of a compact set of a circle group rotation

Let $\mathbb{T}$ be the circle group and $R:\mathbb{T\rightarrow T}$ an irrational rotation (hence a minimal system). Suppose there exists a compact set $K\subset\mathbb{T}$ such that for every ...
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56 views

Shift and ergodic measures

Let $X = \{0,1\}^{\mathbb{N}}$ endowed with the product topology and $\sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.$. ...
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1answer
95 views

If $g$ is invariant under an ergodic map then it's almost everywhere constant

Let $(X,B(X),\mu)$ be a probability space, where $X$ is compact metrizable, and $B(X)$ are the Borel sets. Let $f:X\to X$ be a measurable function such that: i) $\forall A\in B(X)$ ...
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0answers
73 views

Birkhoff theorem for irrational rotation

Lately, I have come across this problem, that I was not sure exactly how to tackle. Let $\alpha$ be an irrational number, and let $0 < a < b < 1$. Prove that $\lim_{n \rightarrow \infty} ...
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0answers
35 views

Relationship between conjugacy class and centralizer for measure preserving transformations

Let $(X, \mathcal{B}, \mu)$ be a Lebesgue probability space. Let $\Phi$ be the space of all invertible measure preserving transformations on $(X,\mathcal{B}, \mu )$, endowed with the weak topology. ...
2
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1answer
65 views

Intrinsically Ergodic Factor

Let $(X,T)$ and $(Y,S)$ be two intrinsically ergodic system with the same topological entropy i.e. $\exists ! \mu, \exists ! \nu$ measures of maximal entropy such that ...
4
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1answer
60 views

what is so great about having an invariant measure?

I am a student who just started to learn basic concepts of ergodic theory. It seems like that given a dynamical system, people are very excited to find various invariant measures of the system. But ...
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0answers
49 views

non Lebesgue meausre for the Doubling map

Does there exists invariant non Lebesgue probability measures for the doubling map $T:[0,1)\rightarrow [0,1]$ defined by $ T(x)=2x \,\text{mod}(1)? $ So a probability measure different from ...
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 ...