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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

1611 questions
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Are there infinitely many $n$ such that $2^n$ and $3^n$ both have first digit $7$? (exercise in ergodic theory)

I am in a reading group that is reading Invitation to Ergodic Theory by C.E. Silva. Section 3.2, exercise 10 asks whether there are infinitely many positive integers $n$ such that both $2^n$ and $3^n$ have leading digit $7.$ We think the answer is…
anon
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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 adding real numbers, we have matrix multiplication…
Magnus
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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 $$s_h = \limsup_{N \to \infty} \left | \frac1N \sum_{n = 1}^N \langle x_{n + h}, x_h \rangle \right |.$$ …
JT_NL
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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 entropy. For the case $\alpha =1/2$, I can…
Ivah
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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 (where we allow $(X,\mathfrak{B},m)$ to be $\sigma$-finite)…
user34632
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Ergodic proof of Khinchin-Levy

What is a good reference for the proof of Khinchin-Levy theorem on continued fractions, using the full power of ergodic theory? A google search is not quite yielding the needed stuff.
K. K.
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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 $$\mu(A \cap T^{-n} B) = \mu(A)\mu(B)$$ for all $n \geq N$, then $\mu(A)…
JT_NL
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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 that proof of Von Neumann's Theorem: Birkhoff…
user34632
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If $T^m$ is ergodic, so is $T^{m^2}$?

The HW problem: If $T^m$ is ergodic, show $T^{m^2}$ is ergodic. (Where we can assume $T$ is measure-preserving transformation on a probability space, I think. It wasn't in the problem, but everything we've done has had that hypothesis). So (and this…
Rikimaru
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Why is the map $T(x,y)=(x,xy)$ not ergodic on $S^1\times S^1$.

Here is my reasoning. I use multiplicative notation. Let $f \in L^2$ then it has a Fourier series $$f(x,y)=\sum a_{n,m}x^ny^m$$ because $x^ny^m$ are characters and so they form an orthonormal basis. Now $$f(T(x,y))=\sum a_{n,m}x^{n+m}y^m.$$ If $f$…
Sorfosh
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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 $\mu$ if we have that for any $f \in C(X)$ $$\frac1n…
JT_NL
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Is it known whether 2-mixing continuous systems on a compact metric space are necessarily "pseudo-3-mixing"?

Is it known whether, for every mixing measure-preserving dynamical system $(X,\mathcal{B}(X),\mu,T)$ with $X$ a compact metrisable space and $T$ a continuous map, we have that for all $A,B,C \in \mathcal{B}(X)$, $$ \mu(A \cap T^{-k}(B) \cap…
Julian Newman
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Exercise 2.8.4 in Einsiedler and Ward

$\newcommand{\set}[1]{\\{#1}\\}$ $\newcommand{\ab}[1]{\langle #1\rangle}$ $\newcommand{\mc}{\mathcal}$ $\newcommand{\Z}{\mathbb Z}$ $\newcommand{\C}{\mathbf C}$ Exercise 2.8.4 in Einsiedler and Ward's Ergodic Theory with a View Towards Number Theory…
caffeinemachine
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Product of ergodic system with rotation of cyclic group of finite order.

Let $(X,\mu,T)$ be an ergodic system and $(G,\nu,R)$ be a system with $G=$, a cyclic group of finite order, say $k\in \mathbb{N}$, $R(g^m)=g^{m+1}$ and $\nu(g^m)=\frac{1}{k}$, for $m\in \{0,1,...,k-1\}$. Isn't then the system $(X\times G, \mu…
User
  • 388
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$T(z)=z^n$ preserves haar measure on the unit circle

I'm studying Peter Walter's book Ergodic Theory and I faced an example which is difficult for me to prove: $T(z) = z^n$ preserves Haar measure for $n\in \mathbb{Z}\setminus \{0\} $. I searched on the net and found some useful information here:…
Reza Yaghmaeian
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