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

**32**

votes

**4**answers

3k 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 ...

**32**

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 ...

**28**

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 ...

**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 ...

**25**

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

920 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

**2**answers

656 views

### Distribution of the Error term in GH Hardy's “curious result” $\sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$

In an early paper, GH Hardy talks about the distribution of "curious" sum:
$$ \sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$$
where $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...

**20**

votes

**5**answers

847 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 ...

**19**

votes

**3**answers

980 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 = \frac{1}{\sqrt{n}}\sum_{k=1}^...

**19**

votes

**1**answer

310 views

### Possible limits of $(1/n) \sum_{k=0}^{n-1} e^{i2\pi \cdot 2^k\alpha}$

I made a throwaway comment on math stackexchange the other day that got me thinking about the following question. Let
$$ f_n (\alpha) = \frac1n \sum_{k=0}^{n-1} e(2^k\alpha),$$
where $e(x) = \exp(i2\...

**19**

votes

**1**answer

705 views

### Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...

**18**

votes

**11**answers

4k 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 ...

**18**

votes

**6**answers

2k 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

**1**answer

1k 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 $\sigma$-...

**18**

votes

**0**answers

469 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 ...

**17**

votes

**3**answers

582 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 ...

**14**

votes

**9**answers

4k 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 ...

**14**

votes

**3**answers

1k 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 ...

**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

1k 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

**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 ...

**13**

votes

**2**answers

430 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 ...

**13**

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

455 views

### Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$

I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough.
What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ ...

**12**

votes

**2**answers

585 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

827 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

372 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

687 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 ...

**12**

votes

**2**answers

408 views

### Nuclear operators/spaces and transfer operators

While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...

**12**

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 ...

**12**

votes

**1**answer

488 views

### resampling over Bowen balls

Hello MO World
I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and ...

**11**

votes

**2**answers

925 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 ...

**11**

votes

**3**answers

1k 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 ...

**11**

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 $...

**11**

votes

**6**answers

1k 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 "...

**11**

votes

**1**answer

381 views

### Krein Milman theorem without the axiom of choice

The Krein-Milman theorem asserts that in a locally convex topological vector space, a nonvoid compact convex subset is the closed convex envelope of its extreme points. But I would like to know when ...

**11**

votes

**3**answers

809 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 $T$-...

**11**

votes

**1**answer

435 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

586 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

**1**answer

889 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

**1**answer

255 views

### About positive upper density

For $S\subset \mathbb{N}$ define the upper density as $D^{\ast
}(S)=\limsup_{n\rightarrow \infty }\frac{\left\vert S\cap \{1,2,\ldots,n\} \right\vert }{%
\left\vert n\right\vert }.$
Question: ...

**10**

votes

**3**answers

1k 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 ...

**10**

votes

**2**answers

738 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

**0**answers

401 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 $r$,...

**10**

votes

**0**answers

547 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

**5**answers

1k 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 ...

**9**

votes

**2**answers

637 views

### Is this ergodic inequality true?

Is anything similar to the following inequality true,
$\displaystyle P\{\max_{n \leq k \leq m} |A_k f - A_n f| > \epsilon\} \leq C \frac{||A_m f - A_n f||_1}{\epsilon}$
where $A_n f = \frac{\...

**9**

votes

**2**answers

366 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 $\Omega(...

**9**

votes

**3**answers

685 views

### Examples In Ergodic Theory and Topological Dynamics

I am currently studying basic Ergodic Theory:
Invariant Measures
Poincaré recurrence Theorem
Invariant Measure For Continuous Transformations
The Ergodic Theorems and Applications
Ergodic ...

**9**

votes

**2**answers

573 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 ...