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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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
14 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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28 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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22 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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39 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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23 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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45 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
23 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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13 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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1answer
41 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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29 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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29 views

how that T is ergodic if and only if the only eigenfunctions $f \in L^2(\mu)$ of $U_T$ corresponding to the eigenvalue $1$ are constant functions.

Let $T:X \rightarrow X$ be a measure-preserving transformation. Assume that $(X,\mathcal{B},\mu)$ is a probability space. Show that T is ergodic if and only if the only eigenfunctions $f \in ...
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1answer
27 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
32 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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23 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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31 views

Show that the 2-d skew product preserves the Lebesgue measure

Let $X=\mathbb{T}^2$. The $2$-$d$ torus and $\lambda$ the Lebesgue measure. Let $\alpha \in \mathbb{R}$ and consider the following map: $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2 : T(x,y) = (x ...
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1answer
87 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
25 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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1answer
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
47 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
65 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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39 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
40 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
44 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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3answers
106 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
76 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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49 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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46 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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18 views

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
31 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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25 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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39 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
26 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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68 views

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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1answer
32 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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9 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
39 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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23 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 ...
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3answers
171 views

Help with conditional expectation on the circle

Let $p >1$ a integer, $X = \mathbb{R} / \mathbb{Z}$ and $\mu\colon \mathcal{B}\to [0,1]$ a probability measure on the Borel subsets of $X$ which is $T \colon X \ni x \to (px \text{ mod }1)$ ...
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1answer
35 views

Limes superior of an ergodic and stationary process

Given a stationary and ergodic sequence of real-valued random variables $(X_n)_{n \geq 0}$ and a Borel$(\bar{\mathbb{R}})$-set A, with $\mathbb{P}[X_0 \in A] > 0$, I want to show, that ...
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1answer
37 views

Transformation preserving the Lebesgue measure

Let $f: [0,1] \to [0,1]$ a measurable transformation that preserves the Lebesgue measure $\mu$. So we have that $\liminf_n n\mid{f^n(x)-x}\mid\leq1$ $\mu$-almost everywhere. One trying to solve this ...
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16 views

Why are weak-mixing systems considered “random” and compact systems considered “ordered”?

As I understand it, weak-mixing systems sort of tend to become "orthogonal" to themselves on the long run, and compact systems tend to become almost periodic. How is this related to them being called ...
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1answer
49 views

Assertions about measures with computers

Let's consider the Lebesgue measure ($\mu$) over the closed interval $[0,1]$. As you know, $\mu(\mathbb{Q} \cap [0,1]) = 0$. In other way, as far as I know the computer just can represent accurately ...
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120 views

Question about B. Host paper 'Nombres, normaux entropie, translations'

I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out: Let $X = \mathbb{R}/\mathbb{Z}$, $p > 1$ a integer, $D_n = \{kp^{-n}\colon 0 \leq k < p^n ...
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1answer
55 views

Uniform Wiener-Wintner Theorem - proof

I am looking for proof of uniform version of Wiener-Wintner theorem: Let $(X, \mathcal{A}, \mu, T)$ be an ergodic measure preserving system. For $f \in L^1(\mu)$ which is orthogonal to the ...
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1answer
34 views

Operator induced by continuous function and measures

If $X$ is a compact metric space, and $T:X \rightarrow X$ is continuous map, what would be meant by $T_\ast$ is the operator on measures induced by $T$? Allow $\mu$ to be some Borel regular normed ...
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16 views

Ergodic Versus non-Ergodic Processes

Besides time averaging not carrying over to the ensemble average (in the limit), what are the pros and cons of ergodic and non-ergodic processes? Suppose you were in an engineering situation and you ...
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1answer
108 views

Not uniquely ergodic transformation

Could you teach me an example of NOT uniquely ergodic but ergodic transformation? And when any continuous, measurable, and ergodic transformation on a topological space X is uniquely ergodic, how ...
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1answer
62 views

Question about Pollicott-Yuri's proof of Rudolph theorem

On the book Dynamical System and Ergodic theory by Pollicott and Yuri there is a proof of the Rudolph x2 x3 theorem (page 153). It looks very clean in comparison with the original proof but I didn't ...