Questions tagged [ergodic-theory]

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

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Equidistribution of spheres in $\mathbb{R^2}/\mathbb{Z^2}$

Let $\mathbb{H^2}$ be the hyperbolic upper half place, and let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$ acting on $\mathbb{H^2}$. A proof of the equidistribution of spheres on $\mathbb{H^2/\Gamma}$ ...
A. S.'s user avatar
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Any minimal WAP dynamical system is distal

I'm trying to show that any minimal WAP dynamical system $(X, G)$ is almost periodic. By Ellis's joint continuity theorem, it suffices to show that any minimal WAP system is distal. There are many ...
Andy's user avatar
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23 votes
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Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...
alvarezpaiva's user avatar
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Asymptotically full stationary process

Let $(X_n)_{n \in \mathbb{Z}}$ be a stationary process on a finite set $A$. Say that it is asymptotically full if for every increasing sequence of subsets $B_n \subset A^n$ such that $\dfrac{\#B_n}{\#...
Stéphane Laurent's user avatar
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Equivalence relations that are both not treeable and amenable

Hyperfinite equivalence relations are treeable. For the case of uncountable relations, I was wondering if there is a reference to (or simple proof of) the following statement: Let $E$ be a (possibly ...
Vladimir's user avatar
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Link between Harris recurrence and Ergodicity

Is it possible to obtain Harris recurrent Markov chain from Ergodic chain (in Birkhoff sense) under certain assumption? That is, suppose we know a Markov chain is ergodic (in Birkhoff sense); is it ...
user74016's user avatar
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2 answers
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"Typical" convergence rate for the von Neumann mean ergodic theorem

The von-Neumann theorem states that for a unitary operator $U: {\cal H} \mapsto {\cal H}$, where ${\cal H}$ is a Hilbert space, the following holds: $$ \lim_{N\to \infty} \frac{1}{N} \sum_{n=1}^N U^n ...
Lior Eldar's user avatar
8 votes
2 answers
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An algorithm for Poincare recurrence time

Define the function $[0,+\infty) \rightarrow R$: $$ f = \cos (t) + \cos (\sqrt{2} t) + \cos (\sqrt{3} t) + \cos (\sqrt{5} t ) . $$ I want a number $t $ bigger than $10^7$ such that $$ f(t) > 4 -...
wdlang's user avatar
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Who introduced the concept of topological mixing?

I am writing an introduction and I want to know who introduced the concept of topological mixing?
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Decay of cusps in geometrically finite groups

Let $X=\mathbb{H}^{n}/\Gamma$ be a quotient of hyperbolic space of a geometric finite subgroup. Let $\mu$ be the Bowen-Margulis measure on the unit tangent bundle, and $m$ its projection to $X$. Fix ...
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Koopman representation, weakly compact action, Ozawa Popa

Given a weakly compact action (Ozawa-Popa) of a discrete group $\Gamma$ on p.m space $X$, consider the Koopman representation $\pi$ on $L^2(X)$. Compose this representation with the Calkin projection. ...
Florin Radulescu's user avatar
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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
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Non-ergodic Dye Theorem for orbit equivalent automorphisms

The Dye Theorem states that any two free ergodic p.m.p automorphisms of a standard probability space are orbit-equivalent. Question: Is there a version of the above theorem for non-ergodic ...
Łukasz Grabowski's user avatar
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3 answers
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Uniquely ergodicity and polynomial ergodic average

Let $(X,T)$ be a uniquely ergodic system (here X is compact, T is a continuous map form $X$ to itself), so for any continuous function $f:X\rightarrow\mathbb{R}$ we have for any $x\in X$, the ergodic ...
Siming Tu's user avatar
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Invariant mesures for expanding maps of the circle

Is there any characterization for the support of T-invariant measures? where T is a C¹ expanding map of the circle i.e. T'(x)>Lambda>1 for all x in the circle. I know there are periodic and total ...
chaut's user avatar
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Nonconventional ergodic averages for commuting transformations

Let $S$ and $T$ be commuting measure-preserving transformations of a standard probability space $(X,\mu)$, so $S$ and $T$ define an action of $\mathbb{Z}^2$ on $(X,\mu)$. I am wondering about ...
burtonpeterj's user avatar
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Can real-valued Markov processes with continuous surjective sample paths admit a non-trivial "forward-invariant" set?

I have both a more general question (concerning stopping times), and then a more specific application (as described in the title). Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be ...
Julian Newman's user avatar
2 votes
2 answers
691 views

Density of periodic points and density of periodic measures

There are many results (usually connected to specification-like properties) about density of periodic measures in the space of all invariant ones. However some questions that seem to be easy (at first ...
Dmitry Todorov's user avatar
2 votes
2 answers
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Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS)

Consider the the state of a system at time $n$, $X_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X_0$, so we have $$X_n = A_n \cdots A_1X_0.$...
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Do the Birkhoff averages of a measurable stationary homogeneous Markov process in continuous time "converge to the right limit"?

[I've decided to rewrite the question, to make the essential point clearer.] Let $\,\mathbb{R}^{[0,\infty)}:=\{(x_t)_{t \geq 0} : x_t \in \mathbb{R} \ \, \forall t\}$. We say that a set $Y \subset \...
Julian Newman's user avatar
3 votes
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377 views

Discontinuity of Radon-Nikodym derivative for Patterson-Sullivan measures for word metrics on Gromov hyperbolic groups

Let $\Gamma$ be a Gromov hyperbolic group coming endowed with a word metric coming from some finite generating set. Let $\nu$ be an associated Patterson-Sullivan measure (quasi-conformal density). I ...
Yellow Pig's user avatar
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When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space. Let $d$ a word metric on $\Gamma$ coming from some finite set of generators. My question is: Does there exist a ...
Yellow Pig's user avatar
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Classification of ergodic measures for circle expanding maps

Let us consider the classical self-covering of the circle $S^1=\mathbb{R}/\mathbb{Z}$ given by $$\times_d(x) = dx \mod 1$$ where the degree $d$ is any integer greater than $1$. There are a wealth of ...
Benoît Kloeckner's user avatar
4 votes
1 answer
453 views

Lyapunov exponent for circle diffeomorphisms

Let $f:S^1\to S^1$ be an orientation-preserving circle diffeomorphism with irrational rotation number (see here). Then the system $(S^1,f)$ admits a unique invariant measure, say $\mu_f$. Let $\...
Pengfei's user avatar
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3 votes
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On the decay of correlations of an ergodic sequence over the set $X_{0}=0$

The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It ...
David's user avatar
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1 answer
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invariant measure of uniquely ergodic horocycle flow

Let $S$ be a compact connected orientable surface of variable negative curvature, and let $M=T^1S$ be the unit tangent bundle of $S$. Then, we know from the paper of Brian Marcus (*) that the negative ...
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A question related to metric Diophantine approximation

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that $$ \left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q} $$ has ...
Kurisuto Asutora's user avatar
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Characterizations of an exotic measure on the open sets in the circle $S^{1}$

Suppose that $U\subseteq S^{1}$ is open where $S^{1}=\{z\in\mathbb{Z}:|z|=1\}$. Then define $\mu_{n}(U)=\max_{t\in S^{1}}\frac{1}{n}\cdot|\{k\in\{1,...,n\}|t\cdot e^{\frac{2\pi ik}{n}}\in U\}|$. ...
Joseph Van Name's user avatar
15 votes
1 answer
1k views

Krein Milman theorem without the axiom of choice

The Krein-Milman theorem asserts that in a locally convex topological vector space, a nonvoid compact convex subset is the closed convex envelope of its extreme points. But I would like to know when ...
Paul-Benjamin's user avatar
7 votes
1 answer
295 views

Mean value of Maass forms

Let $X = SL_2(\mathbb{Z}) \backslash \mathbb{H}$ be the modular surface. Consider a basis of $L^2$-normalized Hecke-Maass cusps forms $\phi_j$ on $X$ with $-\Delta$-eigenvalue $\lambda_j$. Hejhal-...
Pig's user avatar
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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 $1$ (...
Ritwik's user avatar
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2 votes
0 answers
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Cesaro mean of products of converging matrices

Let $S$ be a finite set of states. Let $(M_n)$ be a sequence of transitions on $S$; that is, for every natural number $n$, $M_n$ is a non-negative $|S| \times |S|$ matrix whose rows sum up to 1. ...
Eilon's user avatar
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Is Akcoglu's theorem for power bounded positive operators still an open problem?

I am reading Ulrich Krengel's book, Ergodic Theorems; the theorem of Akcoglu's he mentions of is on page 189, theorem 2.5. " If $T$ is a positive contraction in a space $L_p$ with $1<p<\infty$, ...
Alan's user avatar
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5 votes
1 answer
615 views

Characterizing residually amenable groups

Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as well?...
Pablo's user avatar
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8 votes
2 answers
585 views

limiting distribution of the random walk from irrational rotation

Motivation: If I recall correctly, the simple symmetric random walk from i.i.d binary steps converges in distribution to the Wiener measure (if scaled with $a_n = \sqrt{n}$). What I am wondering is ...
Jisang Yoo's user avatar
11 votes
2 answers
1k views

Can ergodic theory help to prove ergodicity of general Markov chain?

I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking ...
Petite Etincelle's user avatar
5 votes
1 answer
929 views

invariant measures of the expanding maps on the circle

I would be very happy to know about original references for the following results; For the expanding map $x \mapsto mx$ on the circle, (with $m$ some integer greater than 1) (1) There exist ...
user20471's user avatar
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0 answers
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Proof that Markov shift is pointwise dual ergodic

I am looking for a reference of the proof that a Markov shift is pointwise dual ergodic, I tried google it but with no success.
Alan's user avatar
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1 vote
1 answer
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Sectional curvature as a Hamiltonian on the Grassmanization of the tangent bundle

Edit: According to the comments to the previous version of this question, I remove my essential errors in the question. I thank the commenters very much. Let $M$ be a n dimensional manifold. ...
Ali Taghavi's user avatar
1 vote
0 answers
95 views

combinatorial ergodicity and promotion

According to J. Propp, T. Roby, and (I believe) others, a cyclic action on a finite set $S$ given by a bijection $\zeta: S \longrightarrow S$ is said to be ${\it ergodic}$ with respect to a statistic ...
A. Leverkuhn's user avatar
13 votes
2 answers
613 views

Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$

I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough. What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ ...
Selim G's user avatar
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3 votes
0 answers
189 views

Topological pressure for subshifts on a countable alphabet

Apologies for asking two similar questions within a week of each other, I had hoped that asking a finite alphabet version of this question would lead to enlightenment but unfortunately it didn't. ...
Tom Kempton's user avatar
5 votes
3 answers
299 views

Approximating Subshifts From Below

I'm looking to understand how to approximate certain countable alphabet subshifts by Markov shifts, and realised that I don't know how to do it even in the finite alphabet case. My guess is that the ...
Tom Kempton's user avatar
1 vote
0 answers
305 views

Properties of a function from its pullback

Edit: I have now removed the duplication previously referred to. Thank you. Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{...
compmath's user avatar
2 votes
1 answer
199 views

Eigenfunction of ergodic skew product fixed by commutator?

Background: Let $(Y, \mathcal{B},\mu,T)$ be an ergodic probability system and let $G$ be a compact metrizable group with compact subgroup $H$. Given a measurable map $\rho:Y \to G$. We may define the ...
Mathemagician's user avatar
6 votes
0 answers
483 views

Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep

Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...
Alin Galatan's user avatar
1 vote
1 answer
168 views

Is it possible to define the density of the logistic map for $x<0$?

Probability density functions (PDF's) have inherent connections to the field of Dynamical Systems. The motivation for this question can be found in: http://www.stat.cmu.edu/~cshalizi/754/2006/notes/...
China-Hong Kong's user avatar
2 votes
1 answer
396 views

Weak Convergence to Lebesgue Measure

I'm trying to understand the proof given by D. Rudolph in his paper "x2 and x3 invariant measures and entropy". I'm particularly trying to undestand the proof of lema 4.4. Let's consider a secuence ...
Jarana's user avatar
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5 votes
0 answers
141 views

Growth in families of trees

I'm hoping that the question below is simple thermodynamic formalism, but I can't quite make it work. Any help would be very welcome. Let $\Sigma:=\{0,1\}^{\mathbb N}$ and let $\Sigma^*$ be the set ...
Tom Kempton's user avatar
3 votes
0 answers
148 views

Uniform bounds of number of integral points on affine varieties

In Duke-Rudnick-Sarnak 93, Density of integer points on affine homogeneous varieties, one of the consequences is the following, Consider the variety $V_{n,k} = \{A \in Mat_n(\mathbb{Z}): det(A) = k\...
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