Questions tagged [ergodic-theory]

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.

Given a measure space $(X,\mathcal B,\mu)$, a measure-preserving transformation on $X$ is a measurable map $T\colon X\to X$ preserving the measure $\mu$. This means that $\mu(T^{—1}A)=\mu(A)$ for all measurable sets $A\subset X$.

Pre-requisites are measure theory and integration theory.

Topics include, but are not restricted to: Poincaré's recurrence theorem, Birkhoff's ergodic theorem, ergodicity, mixing, and other ergodic properties, the relation to symbolic dynamics, including Markov measures and Bernoulli measures, metric entropy and topological entropy, Shannon-McMillan-Breiman's theorem, topological entropy, variational principles, equilibrium and Gibbs measures, relation to hyperbolic dynamics, and smooth ergodic theory.

Further reading-Wikipedia

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Why is the topological pressure called pressure?

Let us consider a compact topological space $X$, and a continuous function $f$ acting on $X$. One of the most important quantities related to such a topological dynamical system is the entropy. For any probability measure $\mu$ on $X$, one can…
D. Thomine
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Intuition behind Khinchin's constant

Khinchin proved that For almost all reals $r$ with continued fraction representation $[a_o; a_1, a_2, \dots ]$ the sequence $K_n = \left(\prod_{i=1}^{n} a_i\right)^{1/n}$ converges to a constant $K$ (Khinchin's constant) independent of…
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Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a topological space $Y$ such that $Y$ is the union of a…
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Why is "h" used for entropy?

Why is the letter "h" (or "H") used to denote entropy in information theory, ergodic theory, and physics (and possibly other places)? Edit: I'm looking for an explanation of the original use of "H". As Ilmari Karonen points out, Shannon got "H" from…
Quinn Culver
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Reference for Ergodic Theory

I am looking for a good introductory book on ergodic theory. Can someone recommend some introductory texts on that?
j.o.
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Calculate $\lim_{n \to\infty}\sqrt[n]{\{\sqrt{2}\}\{2\sqrt{2}\}\{3\sqrt{2}\}\cdots\ \{n\sqrt{2}\} }$

$$\text{Calculate :}\lim_{n \to\infty}\sqrt[n]{\{\sqrt{2}\}\{2\sqrt{2}\}\{3\sqrt{2}\}\cdots\{n\sqrt{2}\} } . $$ Note: Weyl's equidistributed criterion. The following are equivalent: $$x_n\quad\text{is equivalent modulo 1}$$ $$\forall~…
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Kakutani skyscraper is infinite

Karl E. Petersen's book "Ergodic Theory", chapter 2, exercise 9, on page 56 Prove that for any ergodic measure preserving transformation $T:X\rightarrow X$ on a non-atomic probability space $(X, \mathcal B, \mu)$ there exists a set $A$ of positive…
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Definition of measure-preserving: why inverse image?

In the definition of measure-preserving dynamical system, the crucial equation is $$ \mu \left(T^{-1} \left(A\right)\right) = \mu\left(A\right) . $$ Why is it not the seemingly more natural $$ \mu \left(T \left(A\right)\right) = \mu…
S. Kohn
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$(n^2 \alpha \bmod 1)$ 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 $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. To that end we consider the transformation $T:…
Rudy the Reindeer
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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 probability measures? Is it that the important theorems,…
George
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Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?

This question has now been cross-posted at mathoverflow. While working on a variational problem, I have reached to the following question. Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=1$, and let $D \subseteq \mathbb{R}^2$ be the closed unit…
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Non-ergodic measure

Is there an easy way to see that if $\mu$ and $\nu$ are $T$-invariant measures on the same space $X$, and $\mu \neq \nu$, then $\frac{1}{2}(\mu+\nu)$ is NOT ergodic? I know that ergodic measures are the extreme points of the convex set of invariant…
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A strange bijection without fixed points

Is it possible to construct a bijection $F:\mathbb Z \to \mathbb Z$ without fixed points (i.e. such that $F(i)\neq i$ for all $i \in \mathbb Z$) such that for all $j\in\mathbb Z_{\neq 0}$, $$\lim_{n\to \infty}…
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High-School Level Introduction to Dynamical Systems

In one month I'll be giving a talk to motivated high schools students on a topic of my choice from dynamical systems and/or ergodic theory. I'm having trouble coming up with a topic compelling enough to keep their interest, yet elementary enough to…
A Blumenthal
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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 within $\varepsilon$ of an integer, is an $IP^*$…
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