Questions tagged [ergodic-theory]

Question 1

Let $(S, \mathcal{S})$ be some measurable state space, $\Omega := S^{\mathbb{N}_0}$, $X_i$ the coordinate maps, $\mathcal{A} := \sigma(X_0, X_1, \dots)$ and $\theta: \Omega \rightarrow \Omega, (x_0, ...

Question 2

Suppose that $X$ is a compact metric space with metric $d$ and $T:X\to X$ is continuous. Assume that $(X,T)$ is uniquely ergodic with the Borel probabilistic measure $\mu$ and $(X,\mathcal{B}(X),\mu,...

Question 3

Let $M$ be a smooth connected manifold of dimension $\geq 2$, and let $\phi: \mathbb{R} \times M \to M$ be a complete flow on $M$. Suppose there exists a point $x_0 \in M$ whose orbit is dense in $M$, ...

Question 4

Let $X = \{1,...,d\}^{\mathbb{N}}$. Show that every infinite compact shift invariant subset of X has a non-periodic point. My Attempt: Suppose not. So $K\subset Per(\sigma)$, where $\sigma:X\...

Question 5

Suppose that $(X,\Sigma,\mu, T)$ is an ergodic probability space with $T$ measurably invertible . Suppose that $(U_{i})_{i\in N}$ is a sequence of sets in $\Sigma$ with $\lim_{n\to\infty} \mu(U_{n})=...

Question 6

My question came from this paper by Lalley (p.g. 2114) where he showed that the hitting/harmonic measure from finite ranged random walk on free groups is a Gibbs state. Let $(\Lambda_+,\sigma)$ be a ...

Question 7

I'm trying to understand how to get mixing time results when transitions in a chain are idempotent i.e. $P^2=P$ for transition $P$. Following is a simplified example to illustrate my confusion. Toy ...

Question 8

I'm studying the Wiener-Wintner (and related) ergodic theorems, and I've been running into a bit of confussion when passing the result from ergodic systems to non-ergodic ones. In most of the ...

Question 9

Denote by $\mathcal{B}(\mathbb{R}^\mathbb{N})$ the Borel $\sigma$-algebra of $\mathbb{R}^\mathbb{N}$. Let $\varphi:\mathbb{R}^\mathbb{N}\to \mathbb{R}^\mathbb{N}$ be the shift map. That is, $\phi ((...

Question 10

Setup: We have a real $k\times k$ matrix $A$ which has entries in $\{0,1\}$. Assume further that $A$ is irreducible and aperiodic. Consider the associated shift space \begin{equation}X := \left\lbrace ...

Question 11

Let $\alpha$ be an irrational real number and let $f:\mathbf{N} \to \mathbf{N}$ be defined as $$ f(n) = \lceil \alpha (n+1) \rceil - \lceil \alpha n \rceil. $$ Here, as usual, $\lceil \cdot \rceil$ is ...

Question 12

Suppose we have an ergodic volume preserving Anosov diffeomorphisms are ergodic. One interpretation of the Ergodic theorem is that this space is "homogeneous", i.e. almost every orbit will ...

Question 13

Im reading the book Intoduction to Dynamical Systems by Brin and Stuck. Currently im studying the ergodic theory section and to be more precise the result about Anosov Diffeomorphisms. There is this ...

Question 14

Let $(\Omega,\mathcal M,m)$ be a Borel measure space, where $\Omega$ is a metric space. When is it true that a Lipschitz function $f:\Omega\to\Omega$ maps measurable sets to measurable sets? If $\...

Question 15

Let $\mathcal X$ be a standard Borel space and $\Omega=\mathcal X^{\mathbb N}$ with the left shift $\theta$. A compact group $G$ acts on $\Omega$ by measurable homeomorphisms that commute with $\theta$...

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Questions tagged [ergodic-theory]

Family of i.i.d. random variables is ergodic under shift

Prove that $(X,T^k)$ is uniquely ergodic [closed]

Does a dense orbit imply topological transitivity for flows on manifolds?

Compact shift invariant infinite set

Birkhoffs Theorem for moving measure increasing sets.

Gibbs measure is the distribution of a $k$-step Markov chain if the potential depends on the first $k$ coordinates

Understanding the effect of idempotence on mixing in a Markov chain

Ergodicity in the Wiener-Wintner Ergodic Theorem

Proving that the tail $\sigma$-algebra is almost contained in the $\sigma$-algebra of shift-invariant sets.

Exercise (Remark 3) from Parry-Pollicott's `Zeta functions and the periodic orbit structure...'

How can I show that this sequence is aperiodic and is not even eventually-periodic.

Isotropy in Anosov diffeomorphism

Doubts in the proof of Holder continuity of stable distribuitions of Anosov Diffeo

When is a Lipschitz image of a measurable set measurable?

Does $\Lambda(P)=\Delta_G(P)$ hold for stationary ergodic processes?

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