Question about probability spaces $(X,\mathcal B,\mu)$ with a measurable map $T\colon X\to X$ preserving the measure, that is $\mu(T^{—1}A)=\mu(A)$ for all $A$ measurable.

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21 views

Integrability and area-preservation property of maps

Suppose we have a map defined on a plane, $x_{1}=f(x_{0})$, where $x \in \mathbb{R}^{2}$. Assume it is integable: there exists a function $I$ of the phase space variable $x$ such that $I(x)=I(f(x))$. ...
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45 views

Rotation map on $S^1$ preserves measure

I'm having a little trouble understanding following the example in my book as to why the rotation map $R_{\alpha}$ preserves Lebesgue measure. We have $R_{\alpha}([x])=[x+\alpha]$ and $R_{\alpha}^{-...
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41 views

Question about dense trajectory on $k$-dimensional torus under rotation map

Today when I was doing ergodic theory problems I faced with following problem: Assume rotation map on $k$-dimensional torus under $\alpha=(\alpha_1,...,\alpha_n)$ then orbit of all $x$ in $k$-...
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39 views

Computing Lyapunov Exponents

Consider $\mathcal{T}=S^{1}\times D^{2}$, where $S^{1}=[0,1]\mod 1$ and $D^{2}=\{(x,y)\in\mathbb{R}^{2}|x^{2}+y^{2}\le 1\}$. Fix $\lambda\in(0,\frac{1}{2})$ and define the map $F:\mathcal{T}\to\...
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1answer
46 views

Question About Definition of Lyapunov Exponents

I just have a quick question about the definition of Lyapunov exponents. My textbook defines them for a smooth map $f:M\to M$, where $M$ is a smooth manifold. For $x\in M$ and $v\in T_xM$: $\lambda(x,...
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53 views

Understanding Proof of Poincare Recurrence Theorem

I'm trying to follow a proof in my book of the Poincare Recurrence Theorem, but I have three questions about this proof: Theorem Let $(X,\Sigma,\mu$ be a finite measure space, $f:X\to X$ be a ...
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1answer
26 views

Point with a dense trajectory

Let's consider a map $\varphi: [0, 1] \rightarrow [0, 1]$ so that $x \mapsto \{2x \}$. I would like to find a point $x$ so that its trajectory is everywhere dence in $[0,1]$. Firstly, the basic idea ...
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1answer
38 views

Is there is a map from the 3-dimensional ball to itself that does not admit an invariant measure?

Krylov-Bogolybov theorem states that if $X$ is metrizable compact space and $f: X \rightarrow X$ is continous then it admits an invariant Borel probability measure. I would like to build a map $F$ ...
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12 views

Can you think of interesting examples of equidistribtuion of a surface measure supported on a curve?

I am trying to understand equidistribution, and am trying to think of two dimensional examples where a measure is supported on a one dimensional subspace (line/curve/...), but in the limit under the ...
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1answer
101 views

What probability topics can be read without Measure Theory

I know Intermediate Probability Theory and Statistics (distributions, convergence concepts, characteristic functions, etc.) and am pretty good at these. I would love to know more about Probability ...
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72 views

Orbit of transformation on point in measure space returns to subset

Let $X$ be a measure space and $T:X\to X$ a measure preserving transformation. The Poincare recurrence theorem states that for any $T$ and any $A\subset X$ with $\mu(A) > 0$ (we take $\mu(X)=1$) ...
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17 views

Is there a way to find $\limsup$ and $\liminf$ for ergodic processes almost surely?

Let $\{X_k\}$ be an ergodic process. I know that if $f$ is a smooth real valued function then by Birkoff's ergodic theorem, $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n f(X_k)=\mathbb{E}(f(X_1))\mbox{ ...
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56 views

Bakers map has dense orbit.

Let $X=[0,1]^2$, $L$-the Lebesgue measure on the $\sigma$-field of Borel sets. Define the map $T:X \to X$: $$T(x,y)= \begin{cases} (2x,y) \textrm{ for } x \in [0,\frac{1}{2}), y \in [0,1] \\ (2x-1, \...
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1answer
93 views

Definition of topological entropy

What the meaning of the limit that appears in the definition of topological entropy? Let $X$ a compact metric space and $f\colon X\to X$ a continuous function and a subset $K \subset X$. The ...
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1answer
89 views

What is measure theoretical entropy in multidimensional symbolic dynamical systems?

Can any one describe the term entropy used in dynamical systems, and what is roll of entropy in symbolic dynamical systems and please give the brief introduction on measure theoretical entropy?
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116 views

Ergodic components of Markov chain by transition matrix

I would like to find an algorithm for obtaining all ergodic components of a finite Markov chain with discrete time defined by its transition matrix (i.e. ergodic subchains into which the given chain ...
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1answer
39 views

Measure preserving transformation $T([a,b])\subset P$ if $\lambda(P)=\lambda([a,b])$

"Suppose that a measurable subset $P \subset [0,1]$ and the interval $I = [a,b] \subset [0,1]$ are such that $\lambda(P) = \lambda(I)$, where $\lambda$ is the Lebesgue measure on $[0,1]$. Show that ...
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214 views

Show that Brownian motion on the unit circle is exponentially ergodic and has the uniform measure as its invariant distribution.

My search results keep bring up planar Brownian motion on the unit disk. However, I am specifically referring to $e^{jW_{t}} = [\cos(W_t),\sin(W_t)]^{T}$ where $W_t$ is Brownian motion. I am at a ...
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1answer
41 views

A claim in Krengel's book on Ergodic Theorems.

In Krengel's it's argued that the fact that $\exists 0\ne u \in L_\infty$ orthogonal to $(zI-T)L_1$ , where $z$ is a complex number on the unit circle, $|z|=1$, then $T^* u = zu$. I don't understnad ...
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1answer
32 views

Stationarity and Ergodicity vs. Memorylessness

A (discrete) memoryless information source is (usually) defined as a collection of random variables that are independent and identically distributed. My question is, does memorylessness imply ...
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1answer
17 views

A question on measure from Krengel's book on Ergodic Theorems.

Something which I am not sure how is it inferred. On page 307 of the book Ergodic Theorems by Ulrich Krengel, they write that: " $M_l(x) = \mu(\{ z : k_1(x,z)\geq 1/l \} )$. Let $l(x)$ be the ...
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45 views

Is $X(T) = A \sin(\omega_0 t + \Phi)$ mean ergodic?

This is an example of a tutorial but I think has not been solved properly. Please help me! $X(T) = A \sin(\omega_0 t + \Phi)$ $A$ and $\phi$ are independent $A$ is uniformly distributed over ...
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1answer
148 views

Bernoulli product measure

Let $\Omega=\{0,1\}^\mathbb{N}$ and $\mathcal{A}$ the sigma-algebra generated by the cylinders sets $\{w\in\Omega\vert \forall s \in S, w_s=\epsilon_s\}$ with $S\subset\mathbb{N}$ finite and $\...
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28 views

Moser's Twist Theorem for maps with reflection

Suppose I have a simple two dimensional integrable twist map, such as $x_{1}=x_{0}+y_{0}, \quad y_{1}=y_{0}$. Suppose that I perturb it in such a way that Moser's Twist Theorem is satisfied. What ...
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41 views

finding an invariant measure

Are there any methods for finding an (infinite, absolutely continuous with respect to lebesgue measure) invariant measure $d\mu=f(u,v)dudv$ for something like the following? $$ T:Q\to Q, \ Q=\{u^2+v^...
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1answer
43 views

Can I use Birkhoff's Ergodic Theorem for this problem?

I have a stationary process $\{u_n\}$ and I have a function $f:\mathbb{R}^L\to \mathbb{R}^+$. I want to evaluate the following limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...
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1answer
50 views

Does aperiodicity needed for ergodic theorem to hold true?

Given a irreducible and positive recurrent countable state Markov Chain with unique stationary distribution $\pi$, is the following true $$\frac{1}{n}\sum_{i=1}^{n}X_i \to \sum j \pi_j$$ Or, to ...
2
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1answer
66 views

Use of ergodic theory in numerical simulations

Is ergodic theory used in numerical simulations? The kind of application I have in mind is: for $\alpha$ irrational, $( n\alpha \mod 1)_{n \geq 0}$ is equi-distributed on $[0,1]$, and I imagine that ...
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50 views

Krylov-Bogoliubov theorem in discrete and continuous time

Consider the following two similar frameworks (the first one in discrete time, the second one in continuous time) in which I am trying to apply the Krylov-Bogoliubov theorem. 1) (Discrete time.) ...
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36 views

homeomorphism represented by function

I know that if $f$ is representing homeomorphism $T$ and $f^0(x)=x,f^1(x)=f(x),...,f^n(x)=f(f^{n-1}(x))$ for $n\geq1$, then $f^n$ is monotone and $f^n(x+k)=f^n(x)+k$ for every integer $k$, but why ...
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113 views

Fractional part of $n\alpha$ is equidistributed

Let $\alpha$ be an irrational number. Then the sequence $\{\{n\alpha\}\}$ is equidistributed. I am using the following definition of equidistribution. A sequence $\{a_i\}$ is equidistributed if $\...
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1answer
44 views

Infinite closed shift-invariant set of binary sequences with zero entropy?

Does there exist an infinite closed shift-invariant $X \subset \{0,1\}^\Bbb Z$ with zero topological entropy? How to think of an example? Will periodic points of shift$|_X$ have zero entropy?
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59 views

Mean-Square Ergodicity of Certain Quantities?

I apologize in advance for my lack of mathematical knowledge, especially in the field of stochastic processes, but I will try my best to formulate my question in a mathematical way. Is it possible ...
2
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1answer
49 views

measure-preserving transformations are spectrally isomorphic

If $(X_{1}, \mathcal{B}_{1}, m_{1})$ and $(X_{2}, \mathcal{B}_{2}, m_{2})$ are probability spaces together with measure-preserving transformations $T_{1}:X_{1}\to X_{1}$,$T_{2}:X_{2}\to X_{2}$. How ...
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102 views

Properties of join of open covers

I am studying the basics on Ergodic Theory from Fundamentos da Teoria Ergodica by Oliveira & Viana (you can actually find the pdf on their website), particularly, the properties of topological ...
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74 views

Kac Lemma for staionary ergodic processes

Simple question. I have a stationary ergodic process $U$ on a finite alphaet and I want to prove the Kac's Lemma (see Cover and Thomas - Elements of Information Theory (second ed.) page 445). In the ...
3
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1answer
61 views

Subadditive sequence

Let $f:M\to M$ a continuous in a compact metric space. For each $\phi:M\to\mathbb{R}$ and $n\in\mathbb{N}$, define $\phi_n:M\to\mathbb{R}$ by $$ \phi_n =\sum_{i=0}^{n-1}\phi\circ f^i $$ For a non-...
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22 views

Linear affine random dynamical systems - positive Lyapunov index proof check?

Consider the affine random dynamical system $$ X_n = \mathbf A_n X_{n-1} + R_n, $$ starting from an initial non-zero position $X_0$, where $\mathbf A_n\in\mathbb C^{d\times d}$ and $R_n\in\mathbb C^{...
2
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1answer
48 views

If $f(x) \le f(Tx)$ then $f(x)=f(Tx)$ almost everywhere ( $T$ is $\mu$-invariant )

Let $X$ be a probability space with probability $\mu$. Let $T:X\to X$ be a measurable and $\mu$-invariant transformation, i.e $\mu \left(T^{-1}A \right) =\mu A. $ for each measurable subset $A\subset ...
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2answers
67 views

Translation invariant event in percolation theory in $\mathbb{Z}^d$

At my probability theory proseminar I had a speech about the percolation theory and one of the topics I presented was the uniqueness of an infinite open cluster in Bernoulli bond percolation in $\...
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1answer
33 views

Relation between hitting and return time.

Given a dynamical system $(X,\mathcal{B},\mu,T) $ where $X$ is a space, $B$ is borel $\sigma$-algebra, $\mu$ is a probability measure and $T$ is a $\mu$ invariant transformation i.e. $\mu(T^{-1}(A)=\...
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1answer
79 views

Shannon's definition of ergodicity

In A Mathematical Theory of Communication (1948) Shannon gives a definition of ergodicity for a Markov process. In order to be ergodic the directed graph of the process must have the following ...
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1answer
129 views

Baker's map is ergodic

Define Baker's map by \begin{align} f(x,y) = \begin{cases} (2x,y/2) & \mbox{ if } (x,y)\in[0,1/2]\times[0,1] \\ (2x-1,y/2+1/2) & \mbox{ if } (x,y)\in[1/2,1]\times[0,1] \\ \end{cases} \end{...
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1answer
194 views

Definition of Lebesgue measure on the Circle $X=\mathbb{R}/\mathbb{Z}$.

I was given the following definition: I feel this is not adequate and would only define Lebesgue on $[0,1]$? (Wikipedia uses a pushforard of the complex exponential) Also later I am given a proof ...
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1answer
80 views

Torus translation is ergodic if and only if the components of the translation vector are rationally independent.

I'm reading Ergodic Theory and Differential Dynamics by Ricardo Mane. There is a theorem in the book that states the following: If x $\in$ $R^n$, the translation L $_{\pi(x)}$: $T^n \rightarrow T^n$ ...
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1answer
23 views

Application $\pi$-$\lambda$ lemma one-sided Markov shift

Let $(S_k^{\mathbb{N}},\Sigma_k^{\mathbb{N}},m,\tau)$ be the probability preserving transformation of the one-sided Markov shift, where $\Sigma_k^{\mathbb{N}}$ is the $\sigma$-algebra generated by the ...
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1answer
54 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 (weak-* sense) set of probability measures. $m$ is a probability ...
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1answer
98 views

Angle-doubling map is mixing

Let $$ T:\mathbb{S}^1\to\mathbb{S}^1\\ x\mapsto 2x $$ be the angle-doubling map on the circle. We know that this transformation is ergodic. We want to prove that is mixing. I have to show that $$ ...
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1answer
144 views

Properties of the space of $ T $-invariant probability measures over a compact topological space.

Let $T: X \to X$ be a continuos map defined on the compact space (Maybe Haussdorff) $X$. Denote by $M_T$ the space of $T$-invariant probability measures defined over the Sigma Algebra of Borel sets. ...
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1answer
85 views

Application Birkhoff ergodic theorem

Let $(X,\mathcal{B},m,T)$ be a probability preserving transformation. Let \begin{align*} I:&=\{f\in L^1: f=f\circ T\}\\ B:&=\{g-g\circ T: g\in L^1\} \end{align*} I have to show that $$ \lim_{n\...