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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6
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0answers
109 views

What is Birkhoff's ergodic theorem for $GL_n$?

I was wondering about the general setup of a "noncommutative" ergodic theory. Instead of measurable maps into $\mathbb R$, we should consider measurable maps into $GL_n(\mathbb R)$, and instead of ...
9
votes
1answer
250 views

Why probability measures in ergodic theory?

I just had a look at Walters' introductory book on ergodic theory and was struck that the book always sticks to probability measures. Why is it the case that ergodic theory mainly considers ...
2
votes
1answer
205 views

Convergence of inner product using Cauchy-Schwarz

I'm reading a paper in which the following argument is made (in the proof of Theorem 7). I will try to provide just the essentials necessary to ask my question, in particular omitting the ...
3
votes
1answer
103 views

Equidistribution results vs transcendence degree

Consider $\alpha =(\alpha_1, \dots, \alpha_n) \in \mathbb{R}^n$ linearly independent over $\mathbb{Q}$, then the map $q \mapsto q \alpha:= ( q \alpha_1, \dots, q \alpha_n)$ gives a dense embedding ...
12
votes
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?
5
votes
1answer
340 views

Use the van der Corput Lemma to prove the equidistribution of $\{\alpha n^2\}$

The van der Corput Lemma states Van der Corput Lemma: Let $(x_n)$ be a bounded sequence in a Hilbert space $H$. Define a sequence $(s_n)$ by $$s_h = \limsup_{N \to \infty} \left | \frac1N ...
0
votes
1answer
87 views

Forward orbits and backwards orbits same closures

Assume that $T: X \to X$ is a homeomorphism of a compact metric space to itself, and let $\mu$ be a $T$-invariant Borel probability measure on $X$. I want to show that for almost every $x \in X$ ...
2
votes
2answers
468 views

Importance of Poincaré recurrence theorem? Any example?

Recently I am learning ergodic theory and reading several books about it. Usually Poincaré recurrence theorem is stated and proved before ergodicity and ergodic theorems. But ergodic theorem does not ...
2
votes
1answer
183 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 $\mu$ ...
6
votes
1answer
250 views

Stronger than strong-mixing

I have the following exercise: "Show that if a measure-preserving system $(X, \mathcal B, \mu, T)$ has the property that for any $A,B \in \mathcal B$ there exists $N$ such that $$\mu(A \cap T^{-n} B) ...
5
votes
1answer
163 views

Ergodic Recurrence

My solution concerning a problem about Ergodic Recurrence requires me to prove that $\|P_T 1_B\| > 0$. Where $P_T$ is the projection onto the space $I := \{f \in L^2 : f \circ T = f\}$, $T$ is a ...
3
votes
1answer
396 views

Weyl Equidistribution Theorem and a Limit

At the moment, I'm studying Ergodic Theory and I find myself a little stuck. The Weyl Equidistribution Theorem states that the following are equivalent: 1. For any $f \in L^{1}([0,1])$ and sequence ...
5
votes
1answer
317 views

Uniform mean ergodic theorem

I'm working in Einsiedler and Ward's book on Ergodic Theory and in Exercise 2.5.4 they want to prove the following $$\lim_{N - M \to \infty} \frac{1}{N - M} \sum_{n = M}^{N - 1} U_T^n f \to P_T f.$$ ...
5
votes
1answer
174 views

Estimating $\#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}$ for irrational $\alpha$

Suppose $\{\alpha n\}$ is the fractional part of $\alpha n$. Put $$A_{\alpha}(n) = \#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}.$$ If $\alpha$ is irrational, can I find some constant $K$ such that ...