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
114 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 ...
4
votes
1answer
67 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 ...
2
votes
1answer
63 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 [ ...
2
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0answers
47 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?
2
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1answer
44 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
107 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 ...
0
votes
1answer
47 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 ...
3
votes
0answers
38 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 ...
3
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0answers
84 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 ...
5
votes
1answer
251 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 ...
2
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0answers
46 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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1answer
57 views

Integrability of the supremum of a sum of 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 $$ ...
1
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1answer
64 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 ...
5
votes
1answer
99 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 ...
3
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1answer
59 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 ...
4
votes
1answer
70 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 ...
1
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1answer
26 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
74 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
55 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
47 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 ...
2
votes
1answer
46 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
100 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}: ...
1
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1answer
24 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
46 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. ...
4
votes
2answers
118 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
148 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
votes
1answer
28 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 ...
4
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0answers
95 views

Shift and ergodic measures

Let $X = \{0,1\}^{\mathbb{N}}$ endowed with the product topology and $\sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.$. ...
1
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1answer
199 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) $\forall A\in B(X)$ ...
2
votes
1answer
118 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 ...
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0answers
41 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
votes
1answer
76 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
votes
1answer
76 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 ...
2
votes
0answers
89 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
votes
1answer
74 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 ...
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0answers
89 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
votes
1answer
116 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 ...
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1answer
64 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
68 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
votes
0answers
67 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 ...
3
votes
0answers
77 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 $(1/N)\sum_{j=0}^{N-1} X_j$ ...
2
votes
1answer
114 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
119 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 ...
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1answer
149 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
67 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
127 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 ...
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1answer
230 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
57 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
votes
1answer
78 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
44 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 ...