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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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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1answer
120 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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32 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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23 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
56 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
80 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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1answer
61 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
202 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 ...
3
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3answers
261 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
215 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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74 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). http://mathoverflow.net/questions/145005/poincare-recurrence-theorem-and-convergence-on-compact-metric-...
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59 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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1answer
39 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 and ...
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27 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 http://www....
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1answer
49 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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86 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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28 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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138 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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3answers
193 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
56 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 $\mathbb{P}[...
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1answer
81 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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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
53 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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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
149 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
49 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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1answer
249 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
95 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 ...
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2answers
113 views

Reference for Algebraic Groups in Ergodic Theory

It seems that the theory of algebraic groups is used in ergodic theory. I was hoping someone could recommend an introduction to algebraic groups that assumes a knowledge of commutative algebra ...
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1answer
48 views

Dynamical Systems Question on Definition

I'm working through some old notes on Dynamical systems, and I see a definition that I'm not familiar with. I'll call it property B, because I'm not sure what else to call it. In the notes, we assume ...
2
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1answer
69 views

Questions to the proof of the $L^p$ Ergodic Theorem of Von Neumann

Before giving the Theorem of Von Neumann and asking my questions to its proof, I'll cite the Ergodic theorem of Birkhoff (out of Walters' "An Introduction to Ergodic Theory", p. 34) that is used in ...
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1answer
47 views

Invariant functions on product of ergodic systems is determined by eigenfunctions?

Given an ergodic measure-preserving system $(X,\mathcal{B},\mu,T)$, the product system $(X\times X, T \times T, \mu \times \mu )$ need not be ergodic, in other words: It may have non-trivial invariant ...
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0answers
32 views

Is a discrete random process issued from a sampled continuous ergodic WSS process also ergodic?

I have a continuous time process $\{X_t,t\in\mathbb{R}\}$ that is WSS and ergodic for the 1st and 2nd moments. I create a random discrete process $\{Y_n,n\in\mathbb{N}\}=\{X_t,t=nT\}$ by discretizing ...
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48 views

Mixing System and density argument

A Mixing system is defined as a dynamical system $(\Omega,\phi^t,\mu)$ for which the following relations holds $$ 1)\qquad\lim_{t\rightarrow\infty} \mu(\phi^{-t}(A)\cap B)=\mu(A)\mu(B); $$ $$ 2)\qquad\...
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1answer
46 views

Conditions for “forward” measure-preservation

A transformation $T$ being $\mu$-invariant is by definition a transformation satisfying $$\mu(T^{-1} E) = \mu(E)$$ for all measurable sets $E$. I was wondering what are sufficient conditions for being ...
3
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2answers
85 views

Do Anosov flows exist on two dimensional compact manifolds?

Question: Let $M$ denote a two-dimensional, smooth, compact Riemannian manifold (without boundary). If $\phi:\mathbb{R}\times M \rightarrow M$ is a smooth flow, then prove that $\phi$ is not an ...
3
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1answer
100 views

Ergodic Theorem and flow

In Walters' An Introduction to Ergodic Theory on page 34 the Birkhoff Ergodic Theorem is given as follows: Suppose $T\colon (X,\mathfrak{B},m)\to (X,\mathfrak{B},m)$ is measure-preserving (...
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0answers
51 views

Why is $\left\{T^n x|n\geqslant 0\right\}$ dense in $X$ iff $x\in\bigcap_{n\geqslant 1}\bigcup_{k\geqslant 0}T^{-k}U_n$?

In Walters' An Introduction to Ergodic Theory I found the following Theorem and proof (p. 29): Theorem 1.7. Let $X$ be a compact metric space, $\mathfrak{B}(X)$ the $\sigma$-algebra of Borel ...
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0answers
46 views

A basic doubt on Markov chain/ergodicity

Consider a finite state (no. of state $N$) Markov chain $\{X_n\}$ (all the random variables are bounded) such that there is a state $i*$ such that $$ \sum_{i=1}^{N}p_{ii*}^{(n)} > 0$$ for all $...
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219 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with ...
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1answer
69 views

Ergodic (equivalent characterization)

Let $(\Omega,\mathcal{B},\mu,T)$ be a measuretheoretical dynamical system. Then this system is called ergodic if $$ B\in\mathcal{B}, T^{-1}(B)=B\implies \mu(B)=0\text{ or }\mu(B^C)=0. $$ Show:...
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2answers
71 views

Numerical integration using Birkhoff theorem

There is a method for numerical integration that uses Brikhoff ergodic theorem? For example if we have a irrational number $\alpha$ we know that for every continuous function $f \colon [0,1] \to \...
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1answer
83 views

Existence of ergodic joining

Let $\underline{X}=(X,\mathcal{B},\mu,T)$ and $\underline{Y}=(Y,\mathcal{B},\mu,S)$ be ergodic measure preserving systems on Borel probability spaces. A joining of $\underline{X}$ and $\underline{Y}$ ...
2
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1answer
47 views

Ergodic properties of orthogonal group $O(n)$

The orthogonal group $O(n)$ is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing ...
3
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1answer
97 views

Definition of strong mixing and definition of measure-preserving

I have a few questions regarding measure-preserving dynamical systems $(X,\mathcal{A},\mu,T)$. 1) The definition of measure preserving is always stated as $$\mu(T^{-1}B)=\mu(B),$$ for all $B$. I ...
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1answer
74 views

Unclear inequality in the proof of Birkhoff ergodic theorem.

I'm trying to understand the tricky proof of the ergodic theorem (Birkhoff 1931). My reference is "Ward,Einsiedler - Ergodic theory (with a view towards number Theory)" section 1.6: Consider the ...
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1answer
75 views

Obscure inequality in a passage of a proof (maximal ergodic theorem)

Look at the following excerpt from the book "Einsiedler and Ward- Ergodic Theory, with a view towards Number Theory": I don't understand why the inequality in the red box is valid. Maybe the ...
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222 views

Help understanding a geometric proof of the ergodicity of the Gauss measure for continued fractions

Any $x\in(0,1)$ can be written as a (regular) continued fraction $$ x = \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}} = [a_1,a_2,a_3,\dotsc] $$ An irrational number has a unique expansion, ...
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122 views

Show that the shift map is measurable and measure-preserving

Show that the shift map $\theta$ of Definition 6.3 is measurable and measure-preserving. Not sure how to represent $\theta^{-1}$ which I believe is where I am stopped in solving this problem.