Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

**31**

votes

**2**answers

2k views

### Ergodic Theorem and Nonstandard Analysis

Here is a quote from Lectures on Ergodic Theory by Halmos:
I cannot resist the temptation of
concluding these comments with an
alternative "proof" of the ergodic
theorem. If $f$ is a complex ...

**29**

votes

**4**answers

2k views

### Does anyone know an intuitive proof of the Birkhoff ergodic theorem?

For many standard, well-understood theorems the proofs have been streamlined to the point where you just need to understand the proof once and you remember the general idea forever. At this point I ...

**28**

votes

**4**answers

2k views

### $\exists$ a shot in ideal pocket billiards?

Assume you have one shot with the cue ball in pocket billiards (a.k.a. pool), with
the game idealized in that no spin is placed on the cue ball in
the initial shot, all collisions between billiard ...

**27**

votes

**3**answers

1k views

### Improving a sequence of 1s and -1s

Suppose you take a $\pm 1$ sequence and you want to "improve it" by taking pointwise limits of translates. What properties can you guarantee to get in the limit?
Two examples illustrate what I think ...

**22**

votes

**1**answer

2k views

### Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle

Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
Question: (Furstenberg) Let $\mu$ be a ...

**21**

votes

**2**answers

836 views

### Billiard dynamics for multiple balls

I am interested to learn to what extent results on billiards
in polygons have been extended to multiple balls.
Assume the balls have equal radii and the same mass,
the same initial speed, and all
...

**20**

votes

**5**answers

804 views

### Iterated Circumcircle

Take three noncollinear points (a,b,c), compute the center of their circumcircle x, and replace a random one of a,b,c with x. Repeat. It seems this process may converge to a point, assuming no ...

**18**

votes

**6**answers

1k views

### Quantitative versions of ergodic theorem

Are there any general theorems similar to Birkhoff's ergodic theorem, but giving quantitative estimates on the rate of convergence or average time of recurrence (perhaps with additional assumptions)? ...

**18**

votes

**2**answers

730 views

### Central Limit Theorem(s) for irrational rotation

Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = ...

**17**

votes

**1**answer

984 views

### Generic points and local entropies

Let $X=\{1,\dots,p\}^\mathbb{N}$ be the space of sequences on a finite alphabet with a metric inducing the product topology, and let $\sigma\colon X\to X$ be the shift map. Let $\mu$ be a ...

**15**

votes

**11**answers

3k 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 ...

**14**

votes

**5**answers

2k views

### Category Theory and Ergodic Theory

I am very much interested in finding out about any category theoretical work on dynamical systems and on ergodic theory. On the face of it, it seems that a categorical language can go a long way, at ...

**14**

votes

**1**answer

913 views

### Rokhlin lemma for arbitrary infinite groups.

Let $G$ be an at most countable discrete group acting freely on a standard probability measure space $X$ in a measure preserving way.
It is well known that if $G$ is a finite group then this action ...

**13**

votes

**3**answers

875 views

### Applications of and motivation for von Neumann's mean ergodic theorem

I stated von Neumann's mean ergodic theorem (VNMET) in a talk recently and someone in the audience asked what it was good for. The only application I knew of VNMET was to prove Birkhoff's ergodic ...

**13**

votes

**3**answers

477 views

### Ergodic limits along subsets of $\mathbb{N}.$

Let say that an infinite subsets $A$ of $\mathbb{N}$ is "nice w.r.to ergodic limits", if it can replace $\mathbb{N}$ in the individual ergodic theorem, that is, if it is such that the following ...

**13**

votes

**0**answers

404 views

### Blocking light with mirrored convex objects

There is a long-unsolved problem posed by Janos Pach,
sometimes known as the enchanted forest problem,
which asks if it is possible to block a point light source
in the plane
from reaching
infinity by ...

**12**

votes

**5**answers

2k views

### Proof of Krylov-Bogoliubov Theorem

Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...

**12**

votes

**9**answers

3k views

### Book recommendation for ergodic theory and/or topological dynamics?

Hello,
I'd like to hear your opinion for ergodic theory books which would suit a beginner (with background in measure theory, real analysis and topological groups). I am looking for something well ...

**12**

votes

**2**answers

1k views

### Random walk is to diffusion as self-avoiding random walk is to …?

One can view a random walk as a discrete process whose continuous
analog is diffusion.
For example, discretizing the heat diffusion equation
(in both time and space) leads to random walks.
Is there a ...

**12**

votes

**2**answers

357 views

### Are rounded rectangle billiard dynamics ergodic?

Bunimovich proved that the billiard-ball dynamics in the Bunimovich stadium is ergodic.
(Image from this link.)
Q. Is it known that the ...

**12**

votes

**2**answers

761 views

### Non-integrable ergodic theory

Can anyone help me out with proofs/counterexamples? I'm working on an operator-valued multiplicative ergodic theorem and need what may(?) be a well-known fact. This fact (if true) would help me get ...

**12**

votes

**1**answer

318 views

### Is $\lfloor \log(n!)\rfloor \alpha$ equidistributed on the unit circle?

In this question $\lfloor a\rfloor$ means the greatest integer not exceeding $a$.
Using van der Corput's inequalities one is able to show that $\log(n!)\alpha$ is equidistributed on the unit circle ...

**12**

votes

**3**answers

646 views

### Looking for at least one beautiful and not too technical result in asymptotic group theory

We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this ...

**11**

votes

**1**answer

417 views

### Measure conjugacy and ergodic decomposition

Roughly speaking, this question asks whether there is a measure-conjugacy between two transformations if there are measure-conjugacies between their ergodic components.
Suppose $(X,\mu)$ is a ...

**10**

votes

**2**answers

458 views

### Birkhoff ergodic theorem and the measure of the bad points

In the Birkhoff ergodic theorem we have a PMPS $(X,B,\mu,T)$ and that for any $f\in L^1(X,\mu)$ $\frac{1}{N}\sum_{n=0}^{N-1}f(T^n x)\to \int f \, d\mu,$ in measure, in $L^1$-norm and $\mu$-a.e.
My ...

**10**

votes

**2**answers

749 views

### Connectedness of space of ergodic measures

Let $X = \Sigma_p^+ = \{1,\dots,p\}^\mathbb{N}$ and let $f=\sigma\colon X\to X$ be the shift map. Let $\mathcal{M}$ be the space of Borel $f$-invariant probability measures on $X$ endowed with the ...

**10**

votes

**3**answers

858 views

### Alternative proofs of the Krylov-Bogolioubov theorem

The Krylov-Bogolioubov theorem is a fundamental result in the ergodic theory of dynamical systems which is typically stated as follows: if $T$ is a continuous transformation of a nonempty compact ...

**10**

votes

**3**answers

1k views

### A random walk on random lines

I am wondering if this random walk remains finite with positive probability.
Start with three lines $A,B,C$ that are extensions of an equilateral triangle.
Let $p_0$ be one corner. Generate a line ...

**10**

votes

**1**answer

686 views

### Different uses of the word “ergodic”

There appear to be two definitions of the word ergodic.
The dynamical systems definition says that a measure space $(X,\mathit B, \mu)$ and measure preserving transformation $T: X \mapsto X$ is ...

**10**

votes

**2**answers

562 views

### Invariant measures and recurrent sets.

Suppose $T:X \to X$ is a homeomorphism of a compact metric space. The recurrent set is the set of all points $x \in X$ such that for every $\epsilon>0$ there exists an $n\in \mathbb{Z}$, $n \ne 0$, ...

**10**

votes

**1**answer

1k views

### System with invariant measure, but no ergodic measure.

Question
Examples of continuous transformations $T: X \to X$ such that the family of invariant probability measures $M(T)$ is NOT empty but there is no ergodic measure ($E(T) = \emptyset$).
Notice ...

**10**

votes

**3**answers

740 views

### Supremum amongst Kolmogorov-Sinai entropies: ergodic or just invariant measures.

Cases where
$sup_{\mu \in E(T)} h_\mu(T)
\neq
\sup_{\mu \in M(T)} h_\mu(T)$.
Background
For a topological space $X$,
let $T: X \to X$ be a continuous application.
Then, call the set of ...

**10**

votes

**0**answers

482 views

### Centralizers of group actions

Let a locally compact group $G$ act on a probability space $(X,\mu)$. Define the centralizer by $C(G)=\{\Delta\in Aut(X,\mu)\mid \Delta(gx)=g\Delta(x)\text{ almost everywhere}\}$. $Aut(X,\mu)$ denotes ...

**9**

votes

**2**answers

359 views

### Is the composition of non-wandering maps still non-wandering?

Let $M$ be a compact space and $f,g:M \to M$ whose non-wandering sets satisfy $\Omega(f)=\Omega(g)=M$. Can we have $\Omega(f \circ g)=M$?
Or more specifically, if $\Omega(f)=M$, can we have ...

**9**

votes

**2**answers

312 views

### What time does it take for irrational rotations to hit an interval?

Hi,
Consider $\theta_n = (\theta_0 + n \theta) \mod 1$, $\theta$ being an irrational number, and $\theta_0$ an uniform random variable in $(0,1)$. Is there any estimates for the time it will take ...

**9**

votes

**2**answers

539 views

### Name this periodic tiling

Hello MO,
I've been working on a problem I'm working on in ergodic theory (finding Alpern lemmas for measure-preserving $\mathbb R^d$ actions) and have found some neat tilings, that I presume were ...

**9**

votes

**0**answers

372 views

### Poincaré recurrence and symplectic packings

Question. Is there any example of a path connected symplectic manifold $(M,\omega)$ that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius ...

**8**

votes

**2**answers

658 views

### Reference request: Geodesic flow on a manifold with negative curvature is ergodic

I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result:
The geodesic flow on a manifold with negative curvature is ergodic.
The lecture note that ...

**8**

votes

**5**answers

831 views

### What are the zero entropy invariant measures for an Anosov geodesic flow?

Let $M$ be the double-torus with a hyperbolic Riemannian metric. The geodesic flow on the unit tangent bundle $T^1M$ has many invariant Borel probability measures. In particular there are closed ...

**8**

votes

**1**answer

365 views

### Strange definition of ergodicity

I've already asked this question on math.stack a few days ago and haven't received an answer, so I'm asking here.
In an engineering course, a stationary process was defined to be ergodic if for all ...

**8**

votes

**2**answers

577 views

### Getting unique ergodicity from minimality

It is known that minimality does not imply unique ergodicity (Furstenberg example). I ask whether the implication holds in following particular situation:
Suppose $X$ is a compact space, $f:X \to ...

**8**

votes

**3**answers

483 views

### non-integrable subadditive ergodic theorem

Dear MO_World,
I have (another) question about relaxing the assumptions in the sub-additive ergodic theorem. Apologies if this is something I should know already...
There are a number of statements ...

**8**

votes

**6**answers

900 views

### Examples of transformations which are weak-mixing but not strong-mixing

I was reminded of this topic by some of the answers to this question, where it was noted that "typical" measure-preserving transformations are weak-mixing but not strong-mixing for several senses of ...

**8**

votes

**1**answer

98 views

### Ergodicity of composition with a rotation

Let $T$ be an arbitrary Lebesgue measure-preserving automorphism of the unit interval $I$. Let $R_{\alpha}$ denote rotation by $\alpha$, i.e. $R_{\alpha}(x)=x+\alpha \pmod{1}$ for $x \in I$ and ...

**8**

votes

**3**answers

799 views

### Decomposition of a dynamical system into ergodic componenents

Quick version of the question. Let $(X, \mu)$ be a probability measure space and let $Z$, the group of integers, act on $X$ in a measure preserving way. How can I decompose $X$ into ergodic ...

**8**

votes

**2**answers

430 views

### proofs of ergodicity of Sl(2, Z) action on R^2 without using duality

The group $
G=SL(2, R)$ acts linearly on $\mathbb R^2$. The Lebesgue measure of $\mathbb R^2$ is invariant and ergodic for $G$. There is a proof using duality theorem:
Let $U$ be the upper triangular ...

**8**

votes

**2**answers

728 views

### Fourier transform of x2 invariant measure

Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...

**8**

votes

**2**answers

543 views

### How does the mixing time of a geodesic flow on a surface vary with the genus?

I have been looking at the numerical behavior of a particular quantity (of no direct importance here, though if you must know the gory details start with figure 17 here) associated to the geodesic ...

**8**

votes

**1**answer

240 views

### Random variables invariant under almost automorphisms.

Let $\Omega$ be a standard atomless probability space, we can assume $\Omega=(0,1)$ with Lebesgue measure. A bijection $f:\Omega/A_1\to\Omega/A_2$ is almost automorphism, if $P(A_1)=P(A_2)=0$, $f(A)$ ...

**7**

votes

**6**answers

644 views

### Furstenberg $\times 2 \times 3$ conjecture, bibliography

Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure.
I wanted to have a ...