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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Strongly mixing uniquely ergodic dynamical system

I'm looking for an example of a dynamical system which is both strongly mixing and uniquely ergodic. I've looked around and found lots of discussion of systems which are uniquely ergodic but not ...
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4answers
67 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 ...
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1answer
58 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 \{|\frac{1}{N}\sum_{j=0}^{N-1} ...
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1answer
32 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 ...
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1answer
43 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 ...
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1answer
45 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
37 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
34 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
50 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 ...
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1answer
33 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
32 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 ...
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1answer
26 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
31 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 ...
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1answer
68 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 ...
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1answer
22 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
40 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 ...
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0answers
29 views

Cesaro bounded.

The exercise is from Ulrich Krengel's book, Ergodic Theorems, on pages 173-174. First preliminary notions: a function $h$ with $T^*h=h$ is called harmonic, where $T$ is a contraction in $L_1$. $Y= ...
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47 views

Entropy of beta-expansion

We have the transformation $T: [0,1) \rightarrow [0,1)$ given by $Tx = \beta x \text{ mod } 1$ with $\beta = \frac{1+ \sqrt{5}}{2}$. Calculate the entropy $h_{\mu}(T)$ of $T$ wrt the invariant ...
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29 views

Krengel's ergodic theorem

I am not sure if this is the correct forum for my present question. Returning to it after a rather long period and hence, have forgotten the conventions. I am unable to understand the proof of ...
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58 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
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1answer
28 views

Is the average of a dense orbit ergodic for shift function?

Let $\sigma$ be the shift function in the space of two-sided infinite sequences of $\{0,1\}$, $X=\{0,1\}^\mathbb{Z}$ equipped with product topology. We know that there is some point $x\in X$ with ...
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18 views

Summation methods and entropy

I am aware of the theory of divergent series, but don't know much of it. If you have a text to recommend, I'd be glad to hear it. Suppose I have an infinite-dimensional probability vector $\mathbf{p} ...
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43 views

Do primes modulo k form a normal sequence?

For some $k>2$, form a sequence whose nth term is the nth prime that is not a divisor of $k$ modulo $k$. e.g. for $k=4$ the sequence would be 1,3,1,3,3,1,1,3,3,1,3,1... Is this sequence normal, ...
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32 views

Why Gaussian process is not Ergodic in general?

Can anyone use a simple way to explain this? I heard this in class but I do not know why. By Wiki: a random process is ergodic if its statistical properties can be deduced from a ...
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1answer
32 views

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic. A is an integer matrix such that A has an eigenvalue which is a ...
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1answer
38 views

Showing that Lebesgue measure is preserved by translations of the $d$-dimensional torus

Let $\underline{\alpha}=(\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$. Show that the transformation $R_{\underline{\alpha}}=\mathbb{T}^d \rightarrow \mathbb{T}^d$ defined by ...
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25 views

Transitive Actions, Primitive Actions, and Ergodicity

A group action is transitive iff it has one orbit. Intuitively, this seems to say $G$ shuffles around all the elements of the $G$-set. A group action is primitive iff it has no nontrivial blocks, ...
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94 views

The measure generated by the Cantor staircase and the intersection of the Cantor set with its translate

Suppose that $T$ is the shift $\bmod 1$ of the Cantor set by an irrational number $\alpha\in (0,1)$. Consider the measure $\mu$ on the interval $[0,1]$ generated by the Cantor staircase. I'd like to ...
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1answer
28 views

shift power mod 1 of the cantor set by an irrational number and their intersections

Let $C$ be the Cantor ternary set and consider the shift $T_a$ mod 1 of the interval $[0,1]$ for an irrational number $a\in[0,1]$. I'm wondering whether $T_a^k(C)\cap T_a^l(C)=\emptyset$, $k,l\in ...
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19 views

A sequence of numbers' question. (From Krengel's book on Ergodic Theorems).

On page 136 of Ergodic Theorem's by Ulrich Krengel, in the proof he sets: $c_n= \sup_j (n)^{-1} \sum_{i=0}^{n-1}x_{i+j}$ then he argues that for any $k,m$ positive integers one has: $$c_{km}\leq c_m$$ ...
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1answer
49 views

Normal in base 10

We consider a decimal expansion $x=\sum^{\infty}_{i=1} \frac{d_i}{10^i}$ for $x \in [0,1)$. This expansion is generated by map $Tx=10x ($mod $ 1)$ defined on $([0,1), \cal B,\lambda)$ with $\lambda$ ...
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1answer
67 views

Induced sytem ergodic implies normal sytem ergodic

Okay, we consider a measure preserving system $(X, \mathcal F, \mu, T)$ and let $A \in\mathcal F$ be such that $\mu(A) > 0$ and $\mu ( \cup ^{\infty}_{n=1} T^{-n}A) = 1 $. Now I want to show that ...
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44 views

Dense orbits on the 2-torus

For $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ and $f: T^2 \rightarrow T^2$ the 2-torus homeomorphism given by $f(x,y) = (x+\alpha, x+y)$. Why is $f$ topologically transitive. If the forward orbit of ...
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1answer
42 views

Ergodic means for an invertible system

Let $(X,B,\mu,T)$ be an invertible dynamicals system (i.e. $T^{-1}$ is measurable and exists almost everywhere) Question 1: is $T^{-1}$ also measure-preserving( $\mu(T(A)=\mu(A)$)? Question 2: if ...
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1answer
51 views

Product of ergodic transformations

I'm asked to give an example, that the product of two ergodic systems is not ergodic in general. I know that for $X_1=X_2=(S^1,B,m,R_a)$ (the irrational rotation on the unit circle with Lebesgue ...
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111 views

Entropy of convolution of measures

Let $G$ be a countable, discrete group, and let $\mu_1,\mu_2$ be probability measures on the group $G$. We define the entropy of $\mu_i$ as $H(\mu_i)=\sum\limits_{g \in G}-\mu_i(g)\log(\mu_i(g))$ ...
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2answers
80 views

Ergodic system has a.e. dense orbits

One more question: Let $X$ be a metric space with probability measure $\mu$ and $T\colon X \to X$ ergodic. $\Rightarrow f$.a.e. $x$ the orbit $O_x=\{T^n(x) : n \in Z\}$ is dense in $X$. So I have ...
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53 views

recurrence of a dynamical system on a compact space

I have a question to an exercise which was already posted (but I'm not allowed to comment it). ...
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47 views

Why this two dynamical Systems are not isomorphic?

Given two dynamical Systems on [0,1) with the Borel $\sigma-Algebra$ and the lebegue measure l. $T_a (x) = x + a$ mod1 $T_2 (x) = 2x$ mod1. Show that this two systems are not isomorphic for any ...
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Why does the set $\pi(C)\cap\pi(D)$ have $\gamma$-measure 1?

I have a question concerning the article Ergodic Theory and Linear Differential Equations by R.A. Johnson. My questions concerns the proof of Lemma 2.3 on page 27, namely the statement ...
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1answer
35 views

A basic ergodic question

I know that irrational number can be approximated by p/q and error less than 1/q^2. But I still cannot give a rigorous proof to this problem. And how to show that the difference between the left ...
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26 views

Why is the constant here equal to $a$?

Let $(\Omega,R)$ denote a flow, where $\Omega$ is compact metric space and $\gamma$ is a normalized measure on $\Omega$. I have problems to understand the following passage in ...
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40 views

Quantitative almost sure recurrence

I'm struggling to prove the following result, which is a special case of a quantitative recurrence result which is due to Michael Boshernitzan: Let $(X,d)$ be a compact metric space with finite upper ...
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1answer
29 views

Ergodic theory for flows: Invariant

Here is the definition of an invariant measure that I do know: Let $(X,\Sigma)$ be a measurable space and let $f\colon X\to X$ be measurable. A measure $\mu$ on $(X,\Sigma)$ is saif to be ...
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How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $ M ^ n $ be a Riemannian manifold, $ f: M \rightarrow M $ be ...
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35 views

What is an unique ergodic measure?

In a book I found the following: [...] on spaces of the form $\Omega\times \mathbb {R}^2$, where $\Omega $ carries a unique ergodic measure. What is meant with $\Omega$ carries a unique ...
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13 views

Invariant measure for 1-2x^2 and limit theorem

I have shown that $f(x) = 1-2x^2$ on [-1,1] has an invariant measure equivalent to lebesgue measure via the change of coordinates $h(x) = \sin\pi x/2$. (I.e. $g(x) = h^{-1}( f( h(x)))= 1 -2|x|$ has ...
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2answers
44 views

Conservative Measures under a group action (reference request)

I was reading a paper and the author define the concept of conservative measure: Let $(X,\mathcal{B})$ a measurable space and $G$ a group that acts on $X$ by $$G\times X:(g,x)\mapsto T_g(x)$$ where ...
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39 views

A question of integral from Krengel's book in Ergodic Theorems.

As the picture depicts, I don't understand how did he get the RHS of: $$\int_0^{2X(\omega)} t^{-1} \psi(dt) \leq m(\log^{+} 2X(\omega))^{m-1} \int_{0}^{ 2X(\omega)} t^{-1} dt$$ Presumably it ...
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28 views

Stochastic matrix.

How do I show that a stochastic matrix, which is irreducible and at least one state is recurrent then all the other states are recurrent as well. And that the markov shift is conservative. The first ...