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

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4
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2answers
243 views

“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 ...
8
votes
2answers
297 views

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 ...
6
votes
1answer
332 views

Who introduced the concept of topological mixing?

I am writing an introduction and I want to know who introduced the concept of topological mixing?
6
votes
4answers
604 views

A follow up question related to entropy

For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each ...
4
votes
3answers
189 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 ...
6
votes
1answer
116 views

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 ...
5
votes
0answers
218 views

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. ...
2
votes
1answer
138 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 ...
1
vote
0answers
63 views

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 ...
6
votes
2answers
110 views

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 ...
0
votes
0answers
81 views

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 ...
2
votes
2answers
411 views

Convexity and semicontinuity of the relative entropy function

There are several different definitions of relative entropy, and some of them are not equivalent. Following is the definition we will use in this question. Let $M$ be a closed manifold and ...
9
votes
1answer
120 views

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 ...
1
vote
1answer
47 views

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 ...
3
votes
1answer
171 views

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 ...
1
vote
1answer
292 views

Does specification implies that entropy map is upper semicontinuous?

Let $(X,d)$ be a compact metric space and f a continuous transformation on X. f has the specification if one can always find a single orbit to interpolate between different pieces of orbits, up to a ...
3
votes
1answer
305 views

First integrals of a 3D incompressible flow

Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\nabla\cdot q\equiv 0$ ...
2
votes
0answers
62 views

Do the Birkhoff averages of a measurable stationary homogeneous Markov process in continuous time “converge to the right limit”?

Let $\,(P_x^t)_{x \in \mathbb{R} , t \geq 0}\,$ be a measurable Markovian family of transition probabilities - that is, a family of Borel probability measures $P_x^t$ on $\mathbb{R}$ such that for ...
1
vote
2answers
129 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 ...
1
vote
2answers
86 views

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 ...
6
votes
2answers
403 views

Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...
2
votes
0answers
123 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 ...
1
vote
1answer
162 views

Find a sequence with uniform frequencies and recurrent property

Given any 4 positive numbers $p_{00}\,,p_{01}=p_{10}\,,p_{11}$,such that the sum of the 4 numbers is 1, now I want to find a sequence in $\{0\,,1\}^\mathbb{N}$ such that this sequence has uniform ...
6
votes
0answers
183 views

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 ...
4
votes
1answer
125 views

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 ...
2
votes
1answer
139 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 ...
6
votes
1answer
745 views

Characterization of amenable actions

Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every ...
11
votes
1answer
318 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 ...
3
votes
0answers
118 views

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 ...
2
votes
1answer
127 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 ...
19
votes
3answers
906 views

Central Limit Theorem(s) for irrational rotation

Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = ...
2
votes
0answers
106 views

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 ...
5
votes
0answers
140 views

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 $\mu_{n}(U)=\max_{t\in S^{1}}\frac{1}{n}\cdot|\{k\in\{1,...,n\}|t\cdot e^{\frac{2\pi ik}{n}}\in U\}|$. ...
0
votes
0answers
43 views

Weak convergence of SDE

Let $(X_t,Y_t)$ be the solution to the SDE \begin{equation} \begin{split} dX_t &= f(X_t,Y_t)dt + \sigma_1 X_t dW^1_t\\ dY_t &= g(X_t,Y_t)dt + \sigma_2 Y_t dW^2_t \end{split} \end{equation} ...
6
votes
1answer
171 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 $\lambda_j$. ...
2
votes
0answers
93 views

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 $1$ ...
5
votes
0answers
164 views

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 $1<p<\infty$, ...
1
vote
0answers
88 views

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. ...
17
votes
3answers
558 views

Ergodic limits along subsets of $\mathbb{N}.$

Let say that an infinite subsets $A$ of $\mathbb{N}$ is "nice w.r.to ergodic limits", if it can replace $\mathbb{N}$ in the individual ergodic theorem, that is, if it is such that the following ...
9
votes
3answers
585 views

Examples In Ergodic Theory and Topological Dynamics

I am currently studying basic Ergodic Theory: Invariant Measures Poincaré recurrence Theorem Invariant Measure For Continuous Transformations The Ergodic Theorems and Applications Ergodic ...
8
votes
2answers
210 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 ...
4
votes
1answer
219 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 ...
3
votes
1answer
193 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 ...
3
votes
2answers
337 views

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 ...
5
votes
1answer
338 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 ...
1
vote
1answer
172 views

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. ...
0
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0answers
53 views

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.
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0answers
74 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 ...
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0answers
118 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.$ ...
3
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0answers
74 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. ...