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

learn more… | top users | synonyms

4
votes
0answers
57 views

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 ...
0
votes
1answer
65 views

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 ...
2
votes
0answers
83 views

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 ...
-3
votes
0answers
41 views

invariant mesures and entropy

Fix {0,1}^{N} the Bernoulli space, the shift function S and pick two dynamical systems (S,m1) (S,m2) (m1,m2 S-invariant measures). Let's say you have entropies h1 and h2 for the dynamical systems. ...
0
votes
1answer
46 views

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 ...
4
votes
2answers
275 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
320 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
348 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
1answer
123 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
229 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
2answers
169 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 ...
2
votes
0answers
68 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
132 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
86 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 ...
9
votes
1answer
126 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
49 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 ...
1
vote
2answers
132 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
92 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 ...
2
votes
0answers
82 views

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 ...
2
votes
0answers
132 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 ...
6
votes
0answers
189 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
131 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
147 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 ...
3
votes
0answers
124 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 ...
3
votes
1answer
178 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 ...
2
votes
0answers
113 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
142 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 $\mu_{n}(U)=\max_{t\in S^{1}}\frac{1}{n}\cdot|\{k\in\{1,...,n\}|t\cdot e^{\frac{2\pi ik}{n}}\in U\}|$. ...
11
votes
1answer
324 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 ...
0
votes
0answers
46 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
174 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
98 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 $1$ ...
1
vote
0answers
89 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. ...
5
votes
0answers
170 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 $1<p<\infty$, ...
4
votes
1answer
230 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 ...
8
votes
2answers
217 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 ...
5
votes
1answer
363 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 ...
3
votes
1answer
200 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 ...
0
votes
0answers
56 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.
1
vote
1answer
174 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. ...
1
vote
0answers
75 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 ...
3
votes
0answers
76 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. ...
4
votes
3answers
195 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 ...
1
vote
0answers
121 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.$ ...
2
votes
1answer
130 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 ...
4
votes
0answers
229 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 ...
1
vote
1answer
95 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: ...
-4
votes
1answer
146 views

A question regarding Kingman's theorem as described in “Ergodic Theorems” book written by Ulrich Krengel [closed]

Hi so I am reading Ulrich Krengel's book called Ergodic Theorems. And in the proof of Kingman's theorem I don't understand why on page 48 $$d_n(1,1)\leq d_n(r,r)^{-1/(r-1)}$$, how did he get this ...
2
votes
1answer
140 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 ...
5
votes
0answers
113 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 ...
1
vote
0answers
75 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) = ...