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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12
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10answers
1k views

Reference for Ergodic Theory

I am looking for a good introductory book on ergodic theory. Can someone recommend some introductory texts on that?
8
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1answer
197 views

Showing a Transformation increases measure (Ergodic Theory)

Hoi, ive been breaking my head on this fora few days.. Ive been trying to show that $T:[0,1)\to [0,1)$ given by $$ T(x)= \begin{cases} 3x & \mbox{ if } x\in [0,1/3)\\ ...
5
votes
1answer
369 views

Measure theoretic entropy of General Tent maps

Let $I=[0,1]$ and $\alpha \in ( 0,1)$. Define the Tent map, $T: I \rightarrow I$ by $T(x)= x/\alpha$ for $x \in [0,\alpha]$ and $(1-x)/(1-\alpha)$ for $x \in [\alpha,1]$ Find the measure theoretic ...
4
votes
0answers
123 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
4
votes
1answer
117 views

Prove that $m$ is ergodic.

Let $X$ be a topological space, $f\colon X\rightarrow X$ be a function. Suppose that there exists a unique invariant Borel probability measure $m$. Prove that $m$ is ergodic. Thank you.
2
votes
1answer
38 views

Show: Ergodic implies $f=\text{const a.s.}$

Let $(\Omega,\mathcal{A},\mu,T)$ be a dynamic system in measure theory. Let this system be ergodic. Show then then this implies $$ \forall f\colon\Omega\to\mathbb{R}\mbox{measurable}: ...
8
votes
1answer
293 views

A convergence problem in Banach spaces related to ergodic theory

Suppose $X$ is a Banach space, $T\in B(X)$, satisfied the following condition. $\sup \Big\lVert\frac{1}{n}\sum \limits_{i=0}^{n-1}T^{i}\Big\rVert<\infty$ $\frac{1}{n}\lVert ...
10
votes
1answer
298 views

$(n^2 \alpha \bmod 1)$ is equidistributed in $\mathbb{T}^2$ if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$

I did the following homework question, can you tell me if I have it right? We want to show that the sequence $(n^2 \alpha \bmod 1)$ is equidistributed if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. ...
3
votes
2answers
258 views

$\sigma$-algebra of $\theta$-invariant sets in ergodicity theorem for stationary processes

Applying Birkhoff's ergodic theorem to a stationary process (a stochastic process with invariant transformation $\theta$, the shift-operator - $\theta(x_1, x_2, x_3,\dotsc) = (x_2, x_3, \dotsc)$) one ...
12
votes
1answer
226 views

Why is integer approximation of a function interesting?

I have recently learnt the following result: Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is ...
4
votes
1answer
147 views

Endomorphisms preserve Haar measure

I am having trouble following the argument in page 21 of P. Walters, Intro. to ergodic theory, of the following statement: Any continuous endomorphism on a compact group preserves Haar measure. ...
3
votes
1answer
234 views

The prime numbers do not satisfies Benford's law

A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability, $$ P(d)=\log_{10}(d+1)-\log_{10}d,$$ how do you prove that the prime numbers do ...
3
votes
1answer
144 views

Question about entropy

Let $(X,A,\nu)$ be a probability space and $T\colon X\to X$ a measure preserving transformation $\nu$. Take a measurable partition $P=\{P_0,\dots,P_{k-1}\}$. Let$I$ be a set of all possible ...
2
votes
1answer
162 views

Question about Benford's law

A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability, $$ P(d)=\log_{10}(d+1)-\log_{10}d$$ With Birkhoff's Ergodic Theorem is possible ...
2
votes
1answer
117 views

Space of $T$-invariant probability measures is compact.

I'm trying to show that the space of $T$-invariant probability measures is compact in the weak* topology ($T$ is some measurable transformation from a compact metric space to itself). I'm trying to ...
7
votes
2answers
216 views

What is the distribution of leading digits of the squares?

Inspired by How can 0.149162536... be normal?, I ask for the distribution of the leading coefficients of $1,4,9,16,25,36\ldots$. (namely $1,4,9,1,2,3\ldots$) Does a Benford-like law apply? The online ...
6
votes
2answers
150 views

Recurrent point of continuous transformation in a compact metric space

Given a metric space $(X,d)$ and a transformation $T:X\rightarrow X$, a point $x\in X$ is said to be recurrent iff it belongs to the closure of its orbit $\{T(x), T^2(x),...\}$: more precisely, ...
5
votes
1answer
250 views

Rotation $x \to x+a \pmod 1$ of the circle is Ergodic if and only if $a$ is irrational

I have a book, Ergodic problems of classical mechanics by Arnold/Avez, and in it they prove that rotation $Tx = x+a \pmod 1$ of the circle $M=\{x \pmod 1\}$ is Ergodic if and only if a is irrational. ...
4
votes
1answer
101 views

What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
4
votes
2answers
377 views

Examples: invariant events

In a couple of books I'm reading chapters devoted to the Ergodic theory. As a setting: $(\Omega,\mathcal F,\mathsf P)$ is a probability space, $X:(\Omega,\mathcal F)\to (S,\mathcal S)$ is random ...
3
votes
1answer
51 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 ...
3
votes
1answer
121 views

convergence of entropy and sigma-fields

This question is related to this one. Let $(X_1, X_2, \ldots)$ be a sequence of random variables such that each $X_n$ takes its values in a finite space, say $\{0,1\}$, and the $\sigma$-field ...
2
votes
1answer
42 views

Continuity in the Krylov-Bogoliubov theorem

I have the following proof of the Krylov-Bogoliubov theorem, which asserts that given a compact metric space $X$ endowed with a continuous transformation $T \colon X \to X$ one can find a ...
2
votes
1answer
72 views

Uniform convergence of finitely additive measure along a tree of partitions

Learning about Lebesgue-Rohlin spaces is prominently on my to-do-list, so I'm reading Fundamentals of measurable dynamics by Daniel Rudolph, were I'm stuck on an exercise. Framework There is a ...
2
votes
1answer
179 views

A generic point for a non-ergodic measure

Definition: Let $X$ be a compact metric space and let $\mu$ be a Borel probability measure on $X$, then we say that the sequence $(x_n)$ of elements of $X$ is equidistributed with respect to $\mu$ ...
1
vote
1answer
41 views

Ergodic system, show an implication

Let $(\Omega,\mathfrak{A},\mu,T)$ be a dynamic system in measure theory and $p\geq 1$. Show the implication $(1)\implies (2)$, whereat $$ (1)~~~~~\forall f\in L_{\mu}^p: f= f\circ ...