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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28 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 ...
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23 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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21 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 ...
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0answers
29 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 $O(2^{n-k})$.
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0answers
83 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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0answers
22 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 ...
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1answer
43 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 ...
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0answers
17 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 ...
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1answer
28 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
23 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
20 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. ...
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19 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 ...
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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 ...
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1answer
45 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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0answers
49 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} ...
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1answer
44 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
30 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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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 ...
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1answer
15 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
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 ...
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1answer
44 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
121 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
48 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 $$ ...
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1answer
26 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: $$ ...
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0answers
28 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 ...
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0answers
36 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
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1answer
58 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: $$ ...
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1answer
31 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 ...
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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
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1answer
36 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 $\left(f,\mu\right)$ ...
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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 ...
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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 ...
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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
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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 ...
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0answers
17 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 ...
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0answers
25 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? ...
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1answer
51 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 ...
0
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1answer
18 views

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

The question is: If $\sigma\left(\tau\right)$ is a Borel $\sigma$-algebra such that is genertated by a collection $\mathcal{B}\subseteq\sigma\left(\tau\right)$, then $\mathcal{B}$ is closed under ...
1
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1answer
28 views

¿If a Borel $\sigma$-algebra is generated by a collection of subsets of algebra, then the Borel $\sigma$-algebra is generated by the algebra?

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 algebra $\Gamma$ in $M$ ...
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1answer
34 views

Relation between the density function (measure theory) and density (physics)

I was reading some notes on Ergodic Theory and there is this sentence: Suppose we distribute mass on $X$ according to the mass density $fd\mu$, $f \in L^1(\mu)$,$ f \geq 0$, and then apply $T$ ...
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59 views

How can we construct Markov partitions for Smale Horseshoe and Solenoid?

I was reading the book Ergodic Theory, Hyperbolic Dynamics and Dimension Theory by Luis Barreira. In Chapter 7, it is asked to construct Markov partitions for the Smale Horseshoe and the solenoid. In ...
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34 views

Topologizing ergodic system so that certain function becomes continuous

Let $X$ be a compact metric space and $\mathcal{B}$ its Borel $\sigma$-algebra. Suppose that $(X,\mathcal{B},\mu,T)$ is an invertible ergodic system ($T$ is only a measurable isomorphism, not ...
3
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1answer
70 views

$\mu$ is a $f$-invariant measure

Let $\left(M,\sigma\left(\tau\right),\mu\right)$ a measure space where $\mu$ is a measure finite, $\tau$ is a topology in $M$, i, e, $\sigma\left(\tau\right)$ is a Borel $\sigma-$algebra. Let ...
2
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2answers
70 views

Tent map invariant density

Is there a formula for the invariant density for the tent map $f_t$ (for $\sqrt{2}\leq t\leq 2$)? $$ f_t = \begin{cases} tx, & 0\leq x<1/2 \\ t-tx, & 1/2 \leq x\leq 1. \end{cases} $$ By ...
2
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1answer
51 views

Reference request : Ergodic theory and Number theory

I would like to work on relation between Ergodic theory(Or Dynamical system ) and Number theory but I am looking for a good reference book, Lecture note and Also I like to get familiar with Articles, ...
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26 views

Notation $\mathrm{mod} $ in ergodic theory

Does someone know, what exactly is meant by the following: $$T^{-1}A=A \mod \mu$$ where $\mu$ is a $T$-invariant measure?
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1answer
44 views

Krylov-Bogoliubov Theorem

I've just started to learn for my ergodic theory exam and have the following question, because I can't find anything in my notes: What happens, if the space X is not compact or more specifically what ...
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0answers
30 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
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0answers
37 views

Over a compact space, the set of continuous functions are everywhere dense in the set of all measurable functions

WARNING: The following post certainly contains too much information. I've included the complete context of my question so that it is clear what I'm talking about and where my question comes from. Feel ...
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3answers
65 views

Topologizing Borel space so that certain functions become continuous

Let $X$ and $Y$ be compact metric spaces. Let $f:X \to Y$ be a Borel measurable map and suppose that $T:X \to X$ is a homeomorphism. Can one change the topology on $X$ such that $X$ is still a ...