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.

learn more… | top users | synonyms

2
votes
1answer
22 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 ...
1
vote
0answers
14 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 ...
2
votes
0answers
24 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, \ ...
1
vote
1answer
26 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 $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...
0
votes
1answer
39 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
votes
1answer
26 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 ...
0
votes
0answers
18 views

Reference on the normality of some numbers

I'm searching for a contemporary reference on the normality on basis 10 of the Champernowne and Erdös-Copeland constants, defined as $$ C=0,12345678910111213... $$ $$ E=0,23571113...$$ that is, the ...
-1
votes
0answers
21 views

Topological entropy and degree of smooth mappings

Where can I find the literature "Topological entropy and degree of smooth mappings" by Misiurewicz. Thanks for any help.
1
vote
0answers
35 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.) ...
0
votes
0answers
33 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 ...
-1
votes
0answers
9 views

Ergodic decomposition

What is meant by ergodic-component of a given measure preserving system? I tried to find an answer in Furstenberg book, but it does not contain the answer.
1
vote
0answers
64 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 ...
1
vote
1answer
27 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?
1
vote
0answers
40 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
votes
1answer
39 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 ...
0
votes
0answers
23 views

Ergodic Circle Rotation with Shrinking Forbidden Zone

OK, so, say I have a circle $\mathbb{S} = \mathbb{R}/\mathbb{Z}$. Then, I take my favorite irrational number $\alpha = \log_2(3) \pmod{1}$ and consider the circle rotation $T(x) = x + \alpha ...
0
votes
0answers
34 views

Lower bound on difference of powers of two and three.

Let $k \in \mathbb{N}$ and $n = \lceil k \log_2(3) \rceil$. I seek at tight lower bound on $2^n - 3^k$, where "tight" means faster than $O(2^{n-k})$.
5
votes
0answers
93 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 ...
0
votes
0answers
29 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
votes
1answer
44 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 ...
1
vote
0answers
18 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 ...
1
vote
1answer
31 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 ...
1
vote
2answers
27 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 ...
0
votes
1answer
22 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. ...
0
votes
0answers
20 views

An example of Poincaré recurrence

I must give a lecture on invariant measures, and would like to give nice and simple examples of Poincaré recurrence. For example, for Lebesgue-almost every point in [0,1] such that its decimal ...
0
votes
0answers
12 views

Are binary sequences generated from ergodic maps chaotic?

Chaotic Sequences of IID Binary Random variables with their applications to Communications and related papers by the same Author, Tohru Kohda talks about the statistical properties of symbolic ...
0
votes
1answer
50 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 ...
5
votes
0answers
56 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} ...
0
votes
1answer
53 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 ...
1
vote
1answer
35 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 $_{\pi(x)}$: $T^n \rightarrow T^n$ ...
0
votes
0answers
27 views

Show that in every irreducible and recurrent Markov Chain for all pair of states (i,j) the probability of ever achieving j from i equals 1.

My question is exactly the one in the title. So far I have figured that I can use definition: $$ F_{ij} = P ( \bigcup_{n=1}^{\infty} \{ X_n = j \} | X_0 = i) $$ $$ f_{ij}(n) = P ( X_1 \neq j, \ldots ...
1
vote
1answer
16 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 ...
2
votes
1answer
38 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 ...
1
vote
1answer
48 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 $$ ...
6
votes
1answer
123 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. ...
2
votes
1answer
50 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 $$ ...
2
votes
1answer
27 views

Prove operator is isometry

Let $(X,\mathcal{A},m,T)$ be a probability preserving transformation. Prove that the operator $U:f\mapsto f\circ T$ satisfies $$ \|Uf\|_{p}=\|f\|_{p} $$ for every $1\le p<\infty$. My idea: $$ ...
1
vote
0answers
30 views

T invariant probability measure on a compact space $X$

Let $T:X\to X$ be a continuous map defined on the compact space $X$. I read that if $X$ is a metric space then the set of T-invariant probability measure is not empty. I want to know if this result ...
1
vote
0answers
40 views

Partitioning a space such that a set of ergodic measures is uniquely supported on one of the sets

I'm reading a book on Ergodic theory, and it says that given a set X with a sigma algebra A, and a measurable automorphism T, then you can take a set of ergodic measures $\mu_i$ $\epsilon$ M$_T(X)$ ...
5
votes
1answer
59 views

Beta transformation is Ergodic.

Let $\beta \in \mathbb{R}$ with $\beta >1$. Define $T_{\beta}:[0,1)\to [0,1)$ by: $$T_{\beta}(x)=\beta x-[\beta x]=\{\beta x\} $$. Consider: $$ ...
2
votes
1answer
33 views

Expanding maps of the circle and their coding

A continuous smooth ($C^{10}$-smooth, for example) map $f:S^1\to S^1$ of the circle is called expanding if $\inf_{x\in S^1} f'(x) > 1 $. Here $S^1 = [0,1]/\sim$, the segment with identified ...
1
vote
0answers
18 views

Question about history of Entropy

I have started to study Ergodic theory and entropy by some books and lecture notes more than three months but unfortunately I'm not familiar with history of Entropy (I know some thing about name of ...
2
votes
2answers
49 views

The irrational rotation is ergodic. The proof should use the idea of density point.

Consider $f_{\alpha}:S^{1}\rightarrow S^{1}$ the rotation of unit circle of angle $2\pi\alpha$, and let $\mu$ the Lebesgue measure in $S^{1}$. Let $\alpha$ irrational, show that $\left(f,\mu\right)$ ...
0
votes
1answer
16 views

Example of measurable space generating by a collectión $\mathcal{B}$ such that $\mathcal{B}$ is not colsed under intersections.

There is a $\left(M,\sigma\left(\tau\right)\right)$ meausrable space with $M$ no discrete (**$\sigma\left(\tau\right)$ a Borel $\sigma$-algebra where $\tau$ is a topology in $M$)** such that ...
1
vote
2answers
19 views

If a Borel $\sigma$-algebra is genertated by a collection, then the collection is closed under unions?

The question is: Let $\left(M,\sigma\left(\tau\right),\mu\right)$ a measurable space with $\mu$ a probability. If $\sigma\left(\tau\right)$ is a Borel $\sigma$-algebra such that is genertated by ...
0
votes
0answers
24 views

If $\mu$ is $f$-invariant for a collection generating the Borel $\sigma$-algebra, then $\mu$ is $f$-invariant?

Let $\left(M,\sigma\left(\tau\right)\right)$ a measure space with $\sigma\left(\tau\right)$ a Borel $\sigma$-algebra where $\tau$ is a topology in $M$. Suppose there is a collection $\mathcal{B}$ of ...
0
votes
0answers
42 views

What is $T>0$ large enough such that $\mu\left(B\right)<\varepsilon$?

Let $\left(M,\sigma,\mu\right)$ where $\sigma$ is a Borell $\sigma$-algebra and $\mu$ is a probability $f$-invariant. Let $x\in M$, $E\subset M$ measurable and $f:M\rightarrow M$ a measurable ...
0
votes
0answers
18 views

Rotation on circle ergodic measures

Let $T:\mathbb{S}^1\to\mathbb{S}^1$, $T(x):=x+\alpha\;\;\text{(mod}\,1)$ with $\alpha\in\mathbb{Q}$. Then we know that every $x\in \mathbb{S}^1$ is periodic with period $q$. Show that the measures ...
1
vote
0answers
26 views

Prove a measure is ergodic

Suppose $(X,\mathcal{B})$ be a measurable space, Then assume $T:X\to X$ is uniquely ergodic, i.e. there exists a unique probability invariant measure $m$. Then $m$ is ergodic. Do you have any ideas? ...
3
votes
1answer
53 views

What is an intuitive explanation for Birkhoff's ergodic theorem?

If I'm not familiar with measure theory, what is a good way to understand the idea behind the definitions involved, the interpretation of the theorem, and the proofs thereof? Particularly, it's not ...