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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1answer
25 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
19 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
28 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$_T(X)$ ...
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1answer
35 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
19 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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0answers
15 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
23 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
18 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 ...
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0answers
40 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
42 views

Prove : Unable to understand Relationship between chaos dynamics and $i.i.d$ random variables

I am reading through articles which present the spectral properties of chaotic systems such that they can be candidates for generating pseudo random binary sequences. One such article, is ...
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0answers
12 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
22 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
42 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 ...
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1answer
17 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 ...
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1answer
25 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 $M$ ...
2
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1answer
27 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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0answers
49 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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0answers
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 ...
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1answer
68 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 ...
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2answers
43 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 ...
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1answer
37 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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0answers
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?
3
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1answer
31 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
26 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
32 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
63 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 ...
2
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0answers
41 views

Unique Ergodicity

Show that unique ergodicity is a topological invariant. Is arguing as follows an overkill (hopefully if the logic is correct --- I have a feeling that there has to be a way a T-invariant measure has ...
1
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0answers
20 views

nice stationary process with discrete spectrum

I'd like to have an example of a stationary process $(X_n)_{n \in \mathbb{Z}}$ on a finite alphabet for which the shift $T$ is ergodic and has discrete spectrum, and for which there is a "nice" ...
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0answers
25 views

Positive upper density implies positive upper density of intersection with shift.

The following problem appears in [1]: 2.2.2(a) Use Exercise 2.2.1 to show the following. If $A \subseteq \mathbb{N}$ has positive density, meaning that $\mathbf{d}(A) = \displaystyle \lim_{k ...
2
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1answer
32 views

Almost every $x$ is a cluster point of its own trajectory.

The following problem appears in [1]: 2.3.2(a) Let $(X, d)$ be a compact metric space and let $T:X \rightarrow X$ be a continuous map. Suppose that $\mu$ is a $T$-invariant probability measure ...
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1answer
33 views

How long does it take two identical hidden Markov models run on same observations to forget their initial distributions (if ever)?

Let $H_1$ and $H_2$ be two instances of a finite Hidden Markov Model (HMM) $H$. That is, $H_1$ and $H_2$ have identical state spaces $Q$ as well as identical transition $A$ and emission probabilities ...
4
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1answer
56 views

Strongly mixing uniquely ergodic dynamical system

I'm looking for an example of a dynamical system which is both (measure-theoretically) strongly mixing and uniquely ergodic. I've looked around and found lots of discussion of systems which are ...
2
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4answers
80 views

Can one generate a sequence of natural numbers whose density has a given distribution?

Suppose $\{ p_{k} \}$ is a collection of real numbers with the following properties: 1) $p_k \in (0,1)$ $~~~~$(i.e. $0$ and $1$ are not allowed values) 2) $\sum_{k=1}^{\infty} p_k =1$ An ...
3
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1answer
65 views

Prove existence of Borel set related to the function $f(x)=2x \mod 1$

Let $I=[0,1)$ and $f(x)=2x \mod 1$. Prove that for every $\epsilon>0$ there is $E\subset I$ Borel set s.a $m(I/E)<\epsilon$ and $\lim_{N\to\infty} \sup \left\{|\frac{1}{N}\sum_{j=0}^{N-1} ...
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1answer
41 views

why a minimal dynamical system is a ergodic measure-preserving system?

A dynamical system(DS) is a map $(X,T)$ where $X$ is a compact metric space and $T:X-->X$ is a continuous transformation. A minimal DS means for any point $x$ belongs to $X$, $x$ is a ...
2
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1answer
56 views

if $f$ is weakly mixing then $f^n$ is ergodic?

if $f$ is weakly mixing then $f^n$ is ergodic?I think this is false but I cant find a counter example because I dont know transformations weakly mixing but not mixing.can you prove or give a ...
1
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1answer
48 views

Show that $\lim_{n \rightarrow \infty} \left(\prod_{i=1}^{n} (a_i+1) \right)^{1/n} $ using Birkhoff Ergodic Theorem

Show that for Lebesgue-almost every $x \in [0,1)$, the geometric mean $$\lim_{n \rightarrow \infty} \left(\prod_{i=1}^{n} (a_i+1) \right)^{1/n} $$ exists and has common value. What is this? (no ...
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1answer
46 views

If $x=[a_0,a_1,\dots]$ show that $\mu$-almost every $x \in (0,1/N]$ is infinitely recurrent

Let $G$ be the Gauss map, $$G(x)= \begin{cases} 0 & \text{if} \ x=0 \\ \{\frac{1}{x} \}=\frac{1}{x} \ \mathrm{mod} \ 1 & \text{if $0<x\leq 1$}\end{cases}$$ and $\mu$ be the ...
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2answers
36 views

Show that $ \lim_{n \rightarrow \infty} \frac{|f(T^n(x))|}{n}=0 $

Let $T: (X, \mathcal{A},\mu) \rightarrow (X, \mathcal{A},\mu)$ be ergodic wrt a measure $\mu$ on $(X,\mathcal{A})$. Show that for any $f \in L^1(X,\mathcal{A})$ and $\mu$-almost every $x \in X$ we ...
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1answer
55 views

Show that $Per_n(f)$ of periodic points of period $n$ is finite

Prove that if $f: X \rightarrow X$ is an expansive topological dynamical system of a compact dynamical system $X$, then the set $Per_n(f)$ of periodic points of period $n$ is finite. Any ideas of how ...
1
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1answer
34 views

A polynomial equality for the square of a self-adjoint positive contraction in $L^2$ — from Krengel's book Ergodic theorems

Another mystery from Ulrich Krengel's textbook - Ergodic Theorems (first mystery). This time it's from page 190, in the proof of theorem 2.7. He takes $P=T^2$, where $T$ is a self-adjoint positive ...
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1answer
40 views

Ergodicity of irrational rotation

It is a well-known fact that the irrational rotation on $S^1$ is ergodic with respect to Lebesgue measure. But each proof I have seen uses Fourier Analysis. Now, Can someone give a proof without ...
2
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1answer
27 views

Variation on ergodic estimates

Let the sequence of random variables $\{X_{n}\}, n = 1,2, \ldots$ be a Markov chain, which is sufficiently "Ergodic" so that it has stationary distribution $\pi$ and for a function $f$ the sequence of ...
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1answer
34 views

The map Ti=i+1 mod N is uniquely ergodic

I have a set $X=\{1,2,...,N\}$ and the map $T:X \to X$: $Ti=i+1 \text{ mod } N$. Now I want to show that $T$ is uniquely ergodic and find the unique measure. I know it holds that $T^Nx=x$ iff ...
2
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1answer
76 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
1
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1answer
26 views

Understanding the proof of an Ergodic theorem for Markov chains

An ergodic theorem for Markov chains is as follows. If a Markov chain $(X_n)_{n \ge 0}$ is irreducible and has an invariant distribution $\pi$, then $$\frac{1}{n} \sum_{k=0}^{n-1} f(X_k) \to ...
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1answer
47 views

Lemma 1.7 in chapter 5 in Ergodic Theorems by Ulrich Krengel.

The lemma is on page 181. Here's the lemma and its proof: I don't understand how did they get: $$\lim | S^{n_k+1}f(\omega)|\leq \gamma (\alpha - \epsilon)+(1-\gamma)\alpha$$ The term that is ...