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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14
votes
1answer
456 views

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, ...
12
votes
1answer
223 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 ...
11
votes
10answers
945 views

Reference for Ergodic Theory

I am looking for a good introductory book on ergodic theory. Can someone recommend some introductory texts on that?
11
votes
1answer
165 views

Square root of an integer has only even digits

Is there a non-square positive integer $n$, that $\sqrt{n}$ has only even digits in its decimal representation ?
10
votes
1answer
215 views

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 ...
10
votes
1answer
288 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 $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. ...
10
votes
2answers
304 views

An equivalent condition for strong-mixing

For a measure-preserving (finite) system $(X,\mathcal{B},\mu,T)$, is it correct that the following are equivalent? For every $A,B\in\mathcal{B}$ , $\displaystyle\lim_{n\rightarrow\infty}\mu(A\cap ...
9
votes
1answer
234 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 ...
9
votes
0answers
379 views

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 ...
9
votes
2answers
366 views

Irrational rotation and recurrence time

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the torus, and $\alpha\in(0,1)$ be an irrational number, then the transformation $T$ defined by $Tx=x+\alpha$ is the irrational rotation on $\mathbb{T}$. ...
8
votes
2answers
699 views

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". ...
8
votes
5answers
263 views

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 ...
8
votes
1answer
187 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)\\ ...
8
votes
1answer
288 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 ...
8
votes
1answer
115 views

An ergodic theorem on the circle

Let $S^1$ be a circle (i.e. a closed $1$-dim. manifold) and let $F$ be a non-vanishing smooth vector field on $S^1$. Denote by $(t,x) \mapsto \Phi_t^x$ the flow generated by $F$. I want to show ...
7
votes
2answers
203 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
326 views

Chaos and ergodicity in hamiltonian systems

EDIT : I formerly claimed something incorrect in my question. The Liouville measure needs NOT be ergodic on hypersurfaces of constant energy. Also, I found out that NO hamiltonian system can be ...
6
votes
2answers
136 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, ...
6
votes
1answer
182 views

Lemma in Petersen's *Ergodic Theory*

I'm trying to understand the proof of Lemma 6.2.1 (p.260-261) in Petersen's Ergodic Theory. Specifically, I don't understand why $B_{n}^{A} \in \mathscr{B}(T^{-1}\alpha \vee \dots \vee T^{-n}\alpha)$ ...
6
votes
1answer
244 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) ...
6
votes
0answers
105 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 ...
5
votes
1answer
388 views

Ergodic action of a group

What does it mean and how is it defined if the action of a group is meant to be ergodic? Thank you for your replies!
5
votes
1answer
282 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
157 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 ...
5
votes
1answer
171 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 ...
5
votes
1answer
343 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 ...
5
votes
1answer
102 views

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.
5
votes
1answer
43 views

Historical behavior of the Birkhoff averages

Birkoff Ergodic Theorem: Let be $(X,\mathcal{A},\mu)$ a probability space, $T:X\to X$ a measurable tranformation preserving $\mu$ and $f\in L^{1}(\mu)$, then there exists a $\Sigma \subset X$ and ...
5
votes
1answer
239 views

Pinsker $\sigma$ Algebra

Let $(X,A,\nu)$ be a probability space and $T:X\to X$ a measure-preserving transformation. The Pinsker sigma algebra is defined as the lower sigma algebra that contains all partition P of measurable ...
5
votes
1answer
161 views

A question about continued fractions and Gauss map

For $\alpha \in (0,1)$, write $\alpha$ as a continued fraction like $\alpha=[a_1, a_2, \ldots]$ (note that the implicit $0$th coefficient $a_0=0$ has been omitted), and let $\frac{p_n}{q_n}$ be the ...
5
votes
1answer
333 views

Krylov-Bogoliubov for measurable transformations

In the Krylov-Bogoliubov theorem, the transformation is assumed continuous. Does the theorem hold if the transformation is assumed only to be measurable? If not, what is a counterexample? Edit A few ...
5
votes
1answer
149 views

Entropy of the induced transformation

I need help with this problem: Let $(X,\mathcal{B},\mu,T)$ be a ergodic dynamical system in the probability space $(X,\mathcal{B},\mu)$. Let $A \in \mathcal{B}$ with $\mu(A)>0$. We define the ...
5
votes
1answer
225 views

Ergodic theory in mathematics and physics

How is the theory of ergodic measure-preserving transformations related to ergodicity in the physical sense (which I understood as, very very roughly speaking, that a physical system is called ergodic ...
5
votes
1answer
330 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 ...
5
votes
1answer
78 views

Is there a characterization of the shift-invariant ergodic measures?

Consider probability measures $\mu$ on the space $\{0,1\}^\mathbb{N}$ that are shift-invariant with respect to the left-shift map. Is there a nice characterization of the ergodic shift-invariant ...
5
votes
1answer
75 views

How to check the strong ergodicity of the $SL_2(\mathbb{Z})$-action on the torus?

Suppose $\Gamma\subset SL_2(\mathbb{Z})$ is a non-amenable subgroup, especially, $\Gamma=SL_2(\mathbb{Z})$. Consider the natural action of $\Gamma$ on $S^1\times S^1=T^2$. How to check that this ...
4
votes
3answers
99 views

Showing $\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \log(\cos(\phi))\cos(\phi) \ d\phi = \log(4) - 2 $

This is a minor detail of a proof in 'Chaotic Billiards' by Chernov and Markarian which I foolishly decided to verify. It's page 44 of the book, during the proof that lyapunov exponents exist almost ...
4
votes
1answer
59 views

what is so great about having an invariant measure?

I am a student who just started to learn basic concepts of ergodic theory. It seems like that given a dynamical system, people are very excited to find various invariant measures of the system. But ...
4
votes
1answer
106 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.
4
votes
1answer
255 views

How follows the Strong Law of Large Numbers from Birkhoff's Ergodic Theorem?

We want to prove the strong law of large numbers with Birkhoff's ergodic theorem. Let $X_k$ be an i.i.d. sequence of $\mathcal{L}^1$ random variables. This is a stochastic process with ...
4
votes
1answer
194 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
2answers
64 views

Examples of ergodic geodesic flow

Are there any good examples of a geodesic flow that is ergodic? I know the result that states that the geodesic flow for manifolds with negative curvature are ergodic, but I'm fishing for some ...
4
votes
1answer
89 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
1answer
62 views

It Suffices to Check Mixing on an Algebra

Let $X, \mathcal{A}, \mu$ be a probability space and $T: X \rightarrow X$ a measure preserving measurable map (i.e. $\mu (T^{-1}(A)) = \mu (A)$ for all $A \in \mathcal{A}$). We say $T$ is mixing for ...
4
votes
1answer
521 views

Why are hyperbolic toral automorphisms (e.g. Arnold's cat map) ergodic?

Let $\varphi : \mathbb{T}^2 \to \mathbb{T}^2$ be a hyperbolic automorphism of the torus, induced by a linear map $A : \mathbb{R}^2 \to \mathbb{R}^2$ of determinant $\pm 1$ with no eigenvalues of ...
4
votes
1answer
137 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. ...
4
votes
1answer
241 views

Liouville's theorem: How to get an invariant measure?

Liouville's theorem states that the `natural' 2-from is preserved under the Hamiltonian flow. Apparently this leads to an invariant measure $\mu$ as follows \begin{equation} d\mu = \frac{d\sigma}{|| ...
4
votes
1answer
119 views

Definition of entropy of an ergodic measure

I'm reading a paper in which it is stated that The entropy of an ergodic measure is defined as $$\lim_{n \to \infty} -\frac{1}{n} \sum_{|w|=n} \mu[w] \log \mu[w].\tag{1} \label{eq:1}$$ Here ...
4
votes
1answer
241 views

The Denjoy Theorem

I'm currently studying Denjoy's theorem, which says the following: Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $ \rho$ and the variation of $f^{'}$ (denoted by ...
4
votes
1answer
70 views

Recipe to proof ergodicity?

Consider the following theorem about equivalent formulations of ergodicity. Let $S$ be a measure preserving map on a measrespace $(\Omega,\mathfrak F,\mathbb P)$ and define $$\nu_n(A,\omega) = ...