# Questions tagged [ergodic-theory]

Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

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### Do invariant open sets generate the $\sigma$-algebra of invariant sets?

Let $X$ be a Polish space with Borel $\sigma$-algebra $B(X)$. Let $G$ be a locally compact group. $T:G\times X\to X$ be a continuous action of $G$ on $X$. The $\sigma$-algebra of invariant sets is ...
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### Central limit theorem for irrational rotations

Let $\alpha$ be an algebraic integer of modulus 1, and $R_\alpha z=\alpha z$. Is $$\lim_{n\to\infty}\frac{\log|\sum_{k=1}^n \Re R_\alpha^k z|}{\log n}=\frac12$$ for all $z\in S^1$? Birkhoff's ergodic ...
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### Ergodic theory applied to number theory

I am interested in the links between Ergodic Theory and Number Theory. Can anyone give some references for papers to read in this field? Any open problems? Or ideas where it may be applicable in NT?
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### Rate of convergence for Markov chain in random environment

Let $(\Omega,\mathfrak{F},\mathbb{P})$ be a probability space and $\sigma:\Omega\to\Omega$ be an ergodic, invertible and measure preserving transformation. Consider a family of column stochastic ...
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### Are orbits of a measurable flow always measurable with measure zero?

Let $(X, \mathcal{B})$ be a standard Borel space with a probability measure $\mu$ on $\mathcal{B}$. Let $(T_t)_{t \in \mathbb{R}}$ be a jointly measurable flow (i.e. $(T_t)_{t \in \mathbb{R}}$ is a ...
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### Let $(a_n)_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the set$\{2^n3^m\mid m,n\in N\}$. Find $α$ such that $(a_n)\alpha\pmod1$ is not equidistributed

Let $$(a_n)_{n \in \mathbb{N}} = (1,2,3,4,6,8,9,12,16,18,\cdots)$$ be a sequence that is a listing of the set $$\{2^n3^m \mid m,n \in \mathbb{N}\}$$ We need to find an irrational number $\alpha$ such ...
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### The space of ergodic elements of a topological or Lie group

Let $G$ be a compact topological group with normalized Haar measure $\mu$. An element $g\in G$ is an ergodic element if the mapping $L_g:G \to G$ with $x\mapsto gx$ is an ergodic map. The ...
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### Ruelle's Theorem

When reading the spectral theorem of Ruelle's operator, a crucial question arises: Is there an effective method to compute the leading eigenvalue and the equilibrium state explicitly? Let's consider a ...
1 vote
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### Is it known whether 2-mixing continuous systems on a compact metric space are necessarily "pseudo-3-mixing"?

I asked this question on Math Stack Exchange at https://math.stackexchange.com/questions/4739742/; it received 4 upvotes, but no comments or answers even after a 450-point bounty. The question: Is ...
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### Quantitative version of ergodic theorem in Markov chains

Consider an irreducible Markov chain $X_t$ with finite state space $E$, and unique invariant measure $\pi$. Fix a function $V:E\to\mathbb R$ such that $E_\pi[V]=0$. The ergodic theorem tells us that, ...
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### Ergodicity of linear dynamical systems and convergence of covariance matrices

Let $z(n+1)=Bz(n)+\xi(n+1)$ be an $N$-dimensional linear dynamical system with $\left(\xi(n)\right)_{n\in\mathbb{N}}$ being i.i.d. with $\xi(n)\sim\mathcal{N}(0,\Sigma_{\xi})$. Assumptions: a) The ...
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### the second largest eigenvalue of transfer operators

A Gauss map $T$ is mixing and satisfies Lasota-York inequalities. By Henon's theorem, we know that the transfer operator $\hat{T}$ associated with $T$ has a spectral gap. This means there exists a ...
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### Rotation number for multicomponent Schrödinger equation

Rotation number for Schrödinger equation of the form $$-x''(t) +q(t) x(t) = E x(t)$$ was defined in R. Johnson J. Moser "The rotation number for almost periodic ...
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### When is $f^*:T^*M\to T^*M$ an ergodic map for a diffeomorphism $f:M\to M$?

Let M be a differentiable manifold and $f:M \to M$ be a diffeomorphism. Then $f$ induces a natural map $f^* :T^*M \to T^*M$. The pull back map $f^*$ is a symplectomorphism wrt the ...
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### Spectral disjointness of unitary representations of Type I groups and orthogonality

Background: If $\mathcal{H}$ is a Hilbert space and $U:\mathcal{H}\rightarrow\mathcal{H}$ is a unitary operator, then for each $f \in \mathcal{H}$ the sequence $(\langle U^nf,f\rangle)_{n = 1}^\infty$ ...
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### Nonamenable p.m.p. action on a standard probability space

Let $G$ be a discrete nonamenable countable group acting on a standard probability space $(X,\mu)$ through measure-preserving transformations. Is the action of $G$ always amenable? (Amenable action, ...
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I have recently been interested in going deeper into ergodic theory, beyond an introductory level of knowledge. Background wise, my training has mostly been in stochastic analysis, and I have a ...
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### Proof of Zimmer's cocycle super-rigidity theorem

I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...
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### Proving light escapes mirrors via ergodic theory of billiards

There's a longstanding open problem concerning whether or not it's possible to trap all the light from a point source using a finite collection of circles/lines whose sides are mirrors. This seems ...
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### Mathematical justification for the use of an energy shell in the microcanonical ensemble

I would like to understand an identity used in the deduction of the explicit formula for the probability distribution of the microcanonical ensemble in statistical mechanics. Consider $\Lambda$ to be ...
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### "Ergodic theorem" for Markov kernels

Consider a discrete time Markov chain $(X_t)$ on a finite state space $\mathcal{S}$, with transition matrix $P$. Assume that the chain admits a stationary distribution $\pi$, which I will identify ...
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### A uniform distribution problem coming from higher dimensions

Thinking about an approximation problem related to random walks, the following question came up. Suppose we have $m$ numbers $a_1, \ldots, a_m \in \mathbb{R}$ and that $b \in \mathbb{R}$ is not in the ...
Let $f:X \to X$ be a continuous map, where $X$ is a compact metric space. We say that $f$ is (locally) expanding if there are constants $\lambda >1$ and $\delta_0 > 0$ such that, for all $x, y\... • 118 5 votes 1 answer 223 views ### Difference between the topological entropy and Hausdorff dimension for multifractal formalism I have been reading some results about multifractal formalism. I noticed that some results were proved for the Hausdorff dimension and some results for the topological entropy (in the sense of Bowen). ... • 970 4 votes 1 answer 138 views ### Existence of a domain with simple Dirichlet eigenvalues Let$g$be a smooth Riemannian metric on$\mathbb R^3$that coincides with the Euclidean metric outside a compact set$K$. Does there exist some domain$\Omega$with smooth boundary such that$K \...
Let $X$ be a compact metric space and $T\colon X \to X$ a continuous map. Additionally, let $\mathcal{M}^T(X)$ be the set of $T$-invariant probability measures on $X$ and $\mathcal{E}^T(X)$ the set of ...