Questions tagged [ergodic-theory]

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

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Ergodic transformations, their Poisson suspensions and their Krieger types

Let $T$ be an ergodic nonsingular tranformation of a Lebesgue space. Suppose that the Poisson transformation $T^*$ of $T$ is well-defined and ergodic. Denote by $\alpha$ the Krieger type of $T$ and by ...
Lviv Scottish Book's user avatar
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Correspondence between Hoelder cocycles and Hoelder potential functions for noncompact negatively curved manifolds

Let $\tilde{M}$ be the universal cover of a pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary. When $M$...
Ilya Gekhtman's user avatar
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Random walks on the Poincaré disk

Let $G$ be the group of isometries of the Poincaré disk. Let $\mu$ be a probability measure on $G$, and consider $g_1,..,g_n$ i.i.d. random variables on $G$ distributed according to $\mu$. For $z\in \...
Chevallier's user avatar
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When the Jacobian of unstable measure converges

Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
Adam's user avatar
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Counting simple closed curves

I'm currently trying to understand how to count simple closed curves. I've been reading Alex Wright's survey (https://arxiv.org/pdf/1905.01753.pdf). However, I don't feel like I'm getting the big ...
curious math guy's user avatar
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QI but not ME : not finitely presented groups!

I would like to know the examples of two groups which are not finitely presented and are quasi-isometric (QI), but they are not measured equivalent (ME) (in the sense of Gromov). In the literature, ...
Kajal Das's user avatar
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Hecke equidistribution for $L^1$ functions

Put shortly, the question is, does the Hecke equidistribution hold for $L^1$ functions? To be specific, the version of H.E. I need is $T_N f \rightarrow \int f d\mu$ as $N \rightarrow \infty$, ...
Seungki Kim's user avatar
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Looking for a counterexample for Ruelle's inequality on compact manifold

Let $M$ be a compact differentiable manifold, and $f:M\to M$ be a $C^1$- smooth diffeomorphism. If Assume that $\mu$ be a $f$-invariant probability measure on $M$. Then D.Ruelle proved that $$ h_\...
M.H's user avatar
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Weighted distribution of irrational rotation

Let $\theta\in [0,1]\setminus\mathbb{Q}$. Let $\alpha_0=\theta$ and $\alpha_1=1$. Let $0<p_0<1$ and $p_1=1-p_0$. For a finite word $I=(i_1, i_2, \dots, i_n)\in \{0,1\}^n$, denote by $I'=(i_1, ...
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Powers of ergodic transformations

Here is a lemma that I know to be true, and can prove in half a page or so, but I'm wondering: can anyone supply a reference so that it can simply be quoted in a paper? Lemma Let $T$ be an ergodic ...
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Why do we care about simplicity of the spectrum in Oseledets' theorem?

Oseledets' theorem is a fundamental result in Ergodic theory (see for example here, or Chapter 4 of Lectures on Lyapunov Exponents by Marcelo Viana). The simplicity of the spectrum has been studied ...
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Under what conditions do time averages of ergodic transformations satisfy a central limit theorem?

Let $(X, \mu)$ be a probability space and $T:X\rightarrow X $ an ergodic transformation, i.e. $T$ is measure preserving and the only $T$ invariant subspaces have either measure $0$ or measure $1$ (...
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Any references on infinite-dimensional Fourier-Plancherel theory?

Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...
Alexander Shamov's user avatar
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Reference request: stationary measures as convex combinations of ergodic measures

Does anyone know a good reference for the fact that a stationary probability measure is a convex combination of the stationary and ergodic probability measures? I have found some references for the ...
Jon Peterson's user avatar
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Markov operators and existence of ergodic measures

My question refers to the yesterday's question (see here) of John Learner and goes as follows: Can we deduce the existence of an ergodic measure if we know that an invariant measure exists, but the ...
Almost sure's user avatar
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links and interactions between different approaches to (super-)rigidity

By super-rigidity I mean some theorems concerning the arithmetic subgroups in semi-simple Lie groups. According to Margulis "Discrete subgroups of semi-simple Lie groups" (the book published by ...
genshin's user avatar
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special flows and Rudolph's theorem

The Rudolph's theorem confirm the existence of a special representation of an ergodic flow on the Lebesgue space. (In the book of I.P.Cornfeld entitled Ergodic theory). My question is: what is the ...
Camille Williams's user avatar
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On the Birkhoff ergodic theorem for geodesic flows

Let $S$ be a closed surface endowed with a Riemannian metric of negative curvature and let $US$ be the unit tangent bundle. Let $\mu$ be the Liouville measure on $US$. Let $f: US\rightarrow\mathbb{R}$...
Xin Nie's user avatar
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How Can I Tell when A Subgroup of a Lie Group is Generated by Unipotents?

I'm trying to understand the proof of the Oppenheim conjecture using Ratner's theorem, and I don't immediately see why $SO(2,1)$ is generated by unipotents. Why is $SO(2,1)$ generated by unipotents? ...
Sam Ruth's user avatar
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The definition of simple eigenvalue

This question was posted a long time ago on the mathexchange, but I didn't get any answers there, and despite having discussed it with some colleagues, I don't think I have a definitive answer. I am ...
Mrcrg's user avatar
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Convex combinations of Bernoulli Measures

How big is the weak-* closure of the set of all (finite) convex combinations of Bernoulli measures among all invariant probability measures? I mean, we are in the symbolic space $\{1,2,\ldots,d\}^{\...
Bruno Brogni Uggioni's user avatar
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Rate of convergence of ergodic averages related to irrational rotation

Let $\alpha$ be an irrational number, consider the basic dynamical system $T^{n}(0) = \{n \alpha\}$ where $\{.\}$ denotes the fractional part. Let $a < b$ be two numbers in $[0, 1]$. Then by ...
Mohammad Ali Karami's user avatar
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Questions about some properties of random probabilities and random expectations

Let $(\Omega, \mathcal{A}, \mathbb P)$ be a probability space with $\mathcal{A}$ countably generated, and let $P: \mathcal{A} \times \Omega \to [0,1]$ be a random probability measure. By that I mean $...
aduh's user avatar
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Free ergodic probability measure-preserving actions of the free group

Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $\Gamma$ be a countable group. An action of $\Gamma$ on $X$ is: essentially free if for all $g \in \Gamma \setminus \{e \}$,...
Sebastien Palcoux's user avatar
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390 views

A question about distribution of fractional part of $2^k\alpha$

Let $\{x\}$ be the fractional part of $x$, i.e. $\{x\}=x-[x]$, where $[x]$ is the biggest integer $\leq x$. The question might be well known but I don't know where to look for: Assume $\alpha$ is an ...
Yu Ding's user avatar
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Measure Preserving Transformation Induced by a $*$-automorphism on $L^\infty(X,\mu)$

The following excerpt is from Connes' Noncommutative Geometry Let $(X, \mathcal{B}, \mu)$ be a standard Borel space equipped with a probability measure $\mu$, and let $\ T$ be a Borel ...
Malcolm King's user avatar
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Cesaro bounded Operator which is not power bounded

Good evening! Let X be a banachspace and T a bounded linear operator on X. The cesaro avearges of T are defined as: $A_n:=\frac{1}{n} \sum\limits_{j=0}^{n-1}T^j $ We call T cesaro bounded if: $\...
Matthias's user avatar
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How to detect frequency?

Let $J$ be an arc in $\mathbb{S}^{1}\subset\mathbb{C}$ (no matter open or closed) and $\alpha\in(0,2\pi)$ be an angle such that $\alpha/\pi$ is irrational. Consider in $\mathbb{S}^{1}$ the sequence $...
t22's user avatar
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Does an “almost weakly mixing” transformation admit a non-null ergodic component?

Problem set up: Let $\mathbf X := (X, \mathcal A, \mu)$ be a standard probability space. We say that a measure preserving transformation $T$ on $\mathbf X$ is $\varepsilon$-almost weakly mixing if for ...
Nate River's user avatar
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Substitutions and Sturmian sequences

We know that any substitution can generate sequence, for example the Fibonacci substitution: $\sigma(0)=01, \sigma(1)=0$, then we can define a Sturmian sequence $\omega$, i.e., the fixed point of $\...
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Existence of ergodic measure for measurable maps

The existence of ergodic measures is usually proved under the assumptions that the space $\Omega$ is compact metric and the transformation $T: \Omega \rightarrow \Omega$ is continuous, by using ...
John Learner's user avatar
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Correspondence between fractal sets and trees

In Hillel Furstenberg's series lectures on ergodic theory in fractal geometry, he mentioned his search on finding a one-to-one correspondence between fractal sets and trees, however, I couldn't not ...
Hu Yi Chen's user avatar
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2 answers
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Estimate entropy of a binary process in terms of decay of correlations

Suppose $( X_{n} )$ is an ergodic binary process with $$ \mathbb P(X_{n}=1)= \mathbb P(X_{n}=0)=\frac 12. $$ Naturally the entropy (rate) $h(X)$ of $X=(X_{n})$ satisfies $$ h(X)=\lim_{n\to\infty} \...
user12345678's user avatar
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1 answer
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Accumulation points of the Birkhoff average of $m$

Let $M$ be a closed manifold, $m$ be the normalized volume measure on $M$, and $f:M\to M$ be a $C^2$ transitive Anosov diffeomorphism. Consider the pushforward $f^km$ defined by ----------$f^km(A):=m(...
Pengfei's user avatar
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The Arnold cat map

How can I compute the SRB measure for the cat map? Also any pointers to references for obtaining Markov partitions and recurrence times would be lovely. Thanks
Steve Huntsman's user avatar
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3 answers
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Does mixing automatically imply this seemingly stronger "uniform modulo re-ordering" version of mixing?

THE QUESTION Let $(X,\mathcal{X})$ be a standard Borel space, $T \colon X \to X$ a measurable map, and $\mu$ a $T$-mixing probability measure. Is it necessarily the case that for all $A \in \mathcal{...
Julian Newman's user avatar
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1 answer
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Does full shift have the local product structure?

We say that an invariant measure $\mu$ on some symbolic space $\Sigma$ has local product structure if there is a measurable function $\psi: \Sigma \rightarrow(0, \infty)$ such that the restriction is ...
Adam's user avatar
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3 votes
1 answer
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Does an “almost mixing” transformation admit a non-null ergodic component?

Problem set up: Let $\mathbf X := (X, \mathcal A, \mu)$ be a standard probability space. We say that a measure preserving transformation $T$ on $\mathbf X$ is $\varepsilon$-almost mixing if for every $...
Nate River's user avatar
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3 votes
2 answers
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Can one realize this as an ergodic process?

Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph. We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$ In other words: For ...
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1 answer
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Positive and Null recurrence of Markov Chains on a General State Space

Suppose $X_n$ is an irreducible, aperiodic and Harris recurrent Markov chain. It is well known that in this case, $X_n$ has a stationary distribution $\pi$. Are there any conditions that are ...
joeyg's user avatar
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topological size of the set of weakly mixing measures on the full two-shift

Let $(X, T):= (\{0,1\}^{\mathbb Z}, \text{the shift transformation})$ be the full two shift system. Then the set $M(X, T)$ consisting of all invariant (probability) measures on that system is a ...
Jisang Yoo's user avatar
3 votes
1 answer
412 views

"Strongly mutually singular" families of measures, and the set of ergodic measures

Let $(X,\Sigma)$ be a measurable space [which we can assume to be a standard Borel space if we wish]. Let $\mathcal{S}$ be a set of probability measures on $(X,\Sigma)$. [If we wish, we can assume ...
Julian Newman's user avatar
3 votes
3 answers
351 views

How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...
JustSomeGuy's user avatar
3 votes
1 answer
200 views

Automorphisms of strictly ergodic shift spaces

Let $X$ be a strictly ergodic shift space, and $\omega_1$, $\omega_2$ be two different points in $X$. Is there an automorphism $\Psi$ of $X$ such that $\Psi(\omega_1)=\omega_2$? By an automorphism I ...
Dominik Kwietniak's user avatar
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2 answers
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trivial map on $\sigma-$algebra $\mod{}0$ is trivial

Hi everyone! I am currently studying the basic theory of measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not ...
David Berman's user avatar
3 votes
1 answer
158 views

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, ...
Tiago's user avatar
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1 answer
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Entropy of $f^{m(x)+n}$ of full shift

Let $(X,\mu,f)$ be a two-sided full shift system. Assume that there is $t \in \mathbb{N}$ such that for every $n \in \mathbb{N}$ and $x \in X$, we can define $T(x)=f^{n+m(x)}(x)$, where $m(x) \leq t; $...
Adam's user avatar
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3 votes
1 answer
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Does uniform recurrence imply uniform convergence of the Birkhoff sums?

Let $(X, T, \mathcal F, \mu)$ be an ergodic measure preserving system with finite measure. Suppose $T$ is uniformly recurrent, in the following sense: For every $A \in \mathcal F$, there exists an $M \...
Nate River's user avatar
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3 votes
1 answer
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Extension of Khintchine's recurrence in a simple case

Suppose an ergodic system $(X,\mathcal{B},\mu,T)$ has a Kronecker factor that is isomorphic to an ergodic rotation, say on the Torus. How can one prove that the large intersection property holds for $...
User's user avatar
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1 answer
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Are $C^1$ vector fields generating an ergodic flow $C^0$ dense?

Note: This is a concrete case of the following question: Are almost all measure-preserving flows on compact manifolds ergodic? Let $M$ be a Riemannian manifold with its natural Riemannian measure, and ...
Nate River's user avatar
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