# Tagged Questions

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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### 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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### 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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### 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 [ http://www.cims.nyu.edu/~lsy/...
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### 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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### 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 am ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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. ...
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### 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 $T$-...
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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 \text{... 1answer 133 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 x\... 1answer 76 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 ... 1answer 86 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 f^... 0answers 78 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 \{\... 0answers 81 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 ... 1answer 124 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 ... 2answers 140 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 \... 1answer 177 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 ... 1answer 70 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 ... 1answer 172 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 ... 1answer 269 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 ... 0answers 63 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. ...
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 ...