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

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10
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2answers
555 views

Birkhoff ergodic theorem and the measure of the bad points

In the Birkhoff ergodic theorem we have a PMPS $(X,B,\mu,T)$ and that for any $f\in L^1(X,\mu)$ $\frac{1}{N}\sum_{n=0}^{N-1}f(T^n x)\to \int f \, d\mu,$ in measure, in $L^1$-norm and $\mu$-a.e. My ...
2
votes
1answer
525 views

Liouville's theorem: How to get an invariant measure?

Liouville's theorem states that the `natural' 2-from is preserved under the Hamiltonian flow. Apparently this leads to an invariant measure $\mu$ as follows \begin{equation} d\mu = \frac{d\sigma}{|| ...
6
votes
1answer
139 views

Ergodic Mean for Schrodinger flow

Let us consider the linear Schrödinger equation in $\mathbb{R}^N$ $$ (i\partial _t+\Delta)\,u=0\mbox{ ,}\quad u(0,x)=f$$ with $f\in L^2(\mathbb{R}^N)$, and let $u(t,x)=e^{it\Delta}f$ be its ...
3
votes
0answers
179 views

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 ...
3
votes
1answer
150 views

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 ...
5
votes
1answer
149 views

Is there a survey of recent work relating to the Hausdorff dimension of sets defined through some restriction of digits?

I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through ...
3
votes
1answer
168 views

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 ...
3
votes
1answer
309 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. $\nabla\cdot q\equiv 0$ ...
5
votes
0answers
185 views

Quasicompactness of transfer operators associated to IID matrix products

Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
0
votes
0answers
36 views

Family of random sets represent all integers a.s.

Construct a family of sets $A_n$ such that $$|A_n|=\Theta\left((\log n)^2\right)$$ and the elements of $A_n$ are chosen uniformly at random mod $n$. Say that a set $S$ represents $m\mod{n}$ if there ...
6
votes
1answer
950 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 ...
6
votes
2answers
603 views

Variational Principle for the Entropy

Theorem: Let be $f$ a homeomorphism of a compact metric space $X$, then $$ h_{top}(f)=\sup_{\mu\in \mathcal{M}_{f}}~ h _\mu (f) $$ Question: The above theorem is the famous variational principle ...
6
votes
1answer
139 views

Are irrational multiples of central sets again central?

Let me begin by giving the relevant definitions. A set $A \subset \mathbb{N}$ is said to be central if and only if there exists a topological system $(X,T)$ (with $X$ a compact metric space, $T$ a ...
0
votes
1answer
116 views

order of convergence of the conditional entropy

Let $X_n$ be a random variable distributed on $A_n:=\{1, \ldots, n\}$ and $g_n\colon A_n \to A_n$ such that $\Pr\big(X_n \neq g_n(X_n)\big) \to 0$. Putting $Y_n=g(X_n)$ then by Fano's inequality ...
0
votes
1answer
268 views

Recurrence and transience of cocycle over a dynamical system

Let $X$ be a compact metric space, $T$ a homeomorphism on $X$ and $\mu$ a $T$-invariant probability measure. Let $\phi:X\to\mathbb{R}$ be a continuous function and ...
3
votes
1answer
137 views

The relations between conservative part and conservativity

I revised the question. In smooth ergodic theory, a diffeomorphism is said to be conservative (I), if it preserves the Lebesgue measure. So for some of us, conservativity is just short for ...
8
votes
1answer
248 views

Random variables invariant under almost automorphisms.

Let $\Omega$ be a standard atomless probability space, we can assume $\Omega=(0,1)$ with Lebesgue measure. A bijection $f:\Omega/A_1\to\Omega/A_2$ is almost automorphism, if $P(A_1)=P(A_2)=0$, $f(A)$ ...
2
votes
2answers
533 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 ...
7
votes
2answers
355 views

Silly question about mixing

Let $T$ be a measure-preserving transformation on a probability space $(\Omega,\mathcal B,\mu)$. Assume that for any pair of measurable sets $A,B\in\mathcal B$ with $\mu(A), \mu(B)>0$, one can find ...
5
votes
2answers
269 views

Liverani's CLT (a question)

Let $(\Omega,\mathcal{F},P)$ be a probability space where $\Omega$ is a complete separable metric space, let $T:\Omega\to \Omega$ ` be an ergodic transformation, let $\hat{T}:L^{2}_{_P}(\Omega)\to ...
2
votes
3answers
248 views

The property of a Markov measure

Given $\sigma$ a shift map, $m$ - a Markov measure, $C_a$, $C_b$ - cylinder sets. Suppose $P \in C_b$. The problem is to show the following \begin{equation} m(C_a \cap \sigma^{-1}(P)) = \frac{m(C_a ...
5
votes
1answer
369 views

Non-existence of ergodic measures

Good afternoon. Can anybody give me an example of a continuous map $T:X\to X$ defined on a Polish space $X$ which admits an invariant Borel probability measure but no ergodic Borel probability ...
1
vote
1answer
178 views

power bounded adjacency matrices

A bounded linear operator $T$ on a Banach space $X$ is called power bounded if $\|T^k\|\le M$ for some $M>0$ and all $k\in \mathbb N$. A classical result of Lorch says that if $X$ is reflexive, ...
4
votes
1answer
289 views

Characterising ergodicity of continuous maps

Hello all. Suppose $X$ is a Polish space, $\mu$ is a Borel probability measure on $X$, and $T:X \to X$ is a continuous $\mu$-preserving map which is not ergodic. Does there necessarily exist a Borel ...
0
votes
1answer
182 views

On the affine property of entropy map

Theorem 8.1 in Wlaters' book "An introduction to ergodic theory" says the entropy map is affine. Namely, let $T:X\to X$ be a continuous map of a compact metrc sace. If $\mu, m\in M(X,T)$ and ...
0
votes
1answer
127 views

Ergodicity with respect to the shift

On the space $S=\{ 0,1,\ldots,m \}^{\mathbb{N}}$ for some $m\in \mathbb{Z}_{+}.$ And given a probability $\mu$ on it. Is it true that $\mu$ is fully supported if and only if it is ergodic for the ...
3
votes
1answer
298 views

Invariant measures for Cellular automata

An easy question that I have never been able to answer. Suppose we have the CA on $\{ 0,1,2 \}^{\mathbb{N}}$ with local rule given by $f(x,y)=A_{x,y}$ and $A$ the $3\times 3$ matrix ...
5
votes
2answers
253 views

pointwise ergodic theorem and mean sojourn time

Originally posted on Maths StackExchange, but repositing here because of getting no answer there. Not a research question really - I'm just confused by implications between various ergodic ...
6
votes
0answers
254 views

Do ergodic isometries have discrete spectrum?

Let $X$ be a metric space, $\mu$ a Borel probability measure, and $T:X\rightarrow X$ be an ergodic measure preserving isometry. Is $(X,\mu,T)$ measure theoretically isomorphic to a minimal isometry ...
2
votes
0answers
168 views

Partitions of central sets via dynamical systems

In the book "Recurrence in Ergodic Theory and Combinatorial Number Theory", 1981, Furstenberg introduced the notion of central sets. He proved in Theorem 8.8 that in each finite partition of ...
6
votes
7answers
836 views

Quantization of a classical system (e.g. the case of a billard)

There has been already several questions asking for an introduction to quantum mechanics for a mathematician, but this ons is slightly different, and more restrictive. I know (some) quantum mechanics, ...
5
votes
0answers
383 views

Fibre Mixing for Dynamical Systems

Hi all, I'm interested in understanding a fairly difficult theorem of Lindenstrauss Peres and Schlag. In that paper the authors prove that certain dynamical systems related to beta expansions and ...
5
votes
1answer
283 views

Failure of the Pointwise Ergodic Theorem

It is known that Birkhoff's pointwise ergodic theorem (unlike von Neumann's mean ergodic Theorem) fails to hold for general Folner sequences. The counter-example usually given is the Folner sequence ...
1
vote
2answers
132 views

Relation between entropy of one-parameter group and single elements of this group

My question is motivated by the hypothesis of the Lindenstrauss' proof of arithmetic quantum unique ergodicity, and the answer to my question is certainly known. However, I could not find it in the ...
4
votes
2answers
350 views

Question about entropy

I see this question on the math stack exchange. I found it interesting and still there is no solution there Let $(X,A,\nu)$ be a probability space and $T:X\to X$ a measure preserving ...
9
votes
3answers
630 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 ...
6
votes
2answers
251 views

Minimal period of arithmetic progressions occurring in sets of positive density.

Let $A$ be a subset of ${\mathbb N}$ with positive upper-Banach density, and for each integer $k\geq3$, define $R_k=R_k(A)$ to be the smallest positive integer $r$ such that $A$ contains a length $k$ ...
3
votes
2answers
172 views

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} ...
4
votes
1answer
191 views

Practical way to check for geometric convergence

Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution. When I measure the convergence rate ...
3
votes
3answers
758 views

Connection between properties of Dynamical and Ergodic Systems

Hi All While studying Topological and Ergodic Dynamics, I've got quite preplexed by the different Properties a system might have (minimality, regionally recurring, transitivity, mixing, ergodic, ...
2
votes
1answer
215 views

Absolutely Continuous Invariant Measures for Piecewise Convex Maps

Hi all, I'm interested in a class of 'generalised tent maps' $f:[0,1]\to[0,1]$ for which 1) $f$ is strictly increasing on $[0, \frac{1}{2}]$, $f(0)=0$ and $f(\frac{1}{2})=1$ 2) $f$ is symmetric ...
1
vote
1answer
350 views

Weak convergence, and Cesaro convergence (of mu_n (E) ) imply convergence (of mu_n (E))?

Let $X$ be a compact metric Borel space. Suppose $\mu_{n}(A)\rightarrow\mu(A)$ for all $\mu-$continuity sets $A$ (sets with zero boundary measure), where $\mu_{n}$ is a sequence of probability ...
5
votes
1answer
462 views

Pesin Entropy Formula

In the form that I've seen it stated, the Pesin entropy formula states that if $M$ is a compact Riemannian manifold and $f$ is a $C^{1+\alpha}$ diffeomorphism of $M$ that preserves smooth invariant ...
2
votes
1answer
352 views

Continuity of relative entropy with respect to the weak* topology

Let $X$ be a measurable space, and let $T$ be a measurable transformation $T:X \to X$. Let $\mathcal{P}(X)$ be the space of probability measures on $X$, equipped with the weak* topology. Define the ...
1
vote
1answer
350 views

is the limit of ergodic functions still ergodic?

under what conditions is the limit of a sequence of ergodic functions still ergodic? are there simple counter-examples to this general statement?
13
votes
2answers
418 views

What time does it take for irrational rotations to hit an interval?

Hi, Consider $\theta_n = (\theta_0 + n \theta) \mod 1$, $\theta$ being an irrational number, and $\theta_0$ an uniform random variable in $(0,1)$. Is there any estimates for the time it will take ...
4
votes
1answer
274 views

Stationary, ergodic measures from the structuralist point of view

Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random ...
5
votes
3answers
554 views

Ergodic action of a subgroup

Are there any examples of $H < G$ such that for any pmp ergodic action of the group $G$ on a standard proba space $(X,\mu)$ there exists a set $A$ of $\mu(A)>C$ such that the action of the ...
0
votes
1answer
221 views

Entropy of inverse map for endomorphism case on surfaces

Hi, I know that in the diffeomorphism case the measure entropy of the T:M^{2}-->M^{2} (M smooth Rimannian surface) will be the same as the measure entropy of T^{-1}. But i need to know about the ...
8
votes
2answers
466 views

proofs of ergodicity of Sl(2, Z) action on R^2 without using duality

The group $ G=SL(2, R)$ acts linearly on $\mathbb R^2$. The Lebesgue measure of $\mathbb R^2$ is invariant and ergodic for $G$. There is a proof using duality theorem: Let $U$ be the upper triangular ...