# Questions tagged [ergodic-theory]

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

756 questions
Filter by
Sorted by
Tagged with
5k 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 ...
2k views

### Alternating colors on a line: infinitely often or converge?

Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / …). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...
3k views

### Does there exist 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 ...
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 ...
2k 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 ...
3k 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 ...
2k views

### Who first proved ergodicity of irrational rotations of the circle?

It is a classic result that the irrational rotations of the circle are ergodic. Formally, let $T:\mathbb{T}\to \mathbb{T}$ be defined by $Tz=ze^{2\pi i\alpha}$. If $\alpha$ is irrational, then $T$ is ...
1k views

### For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded? I feel that it is not easy to treat every irrational $x$. I have asked in S.E. ...
714 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 ...
1k 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 ...
9k 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 ...
969 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 ...
1k 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 $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...
4k 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 ...
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)? ...
1k views

6k 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 ...
1k 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 ...
2k 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 ...
668 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 ...
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 $\sigma$-...
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 ...
441 views

### Trapping lightrays with segment mirrors

Q. Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors? I posed this question in several forums before (e.g., here and in an ...
4k 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 ...
1k views

616 views

### Is the following series consisting of equally distributed $\pm 1$ bounded?

Apologise in advance if this problem isn't research-level (I'm quite certain it isn't). It's just I found it quite intriguing because it turned out to be much more subtle than it appeared at my first ...
594 views

### Almost all non-negative real numbers have only finitely many multiples lying in a measurable set with finite measure

Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that Lebesgue measure of $A$ is positive i.e. $0<\lambda(A)<\infty$. Let $S$ be the set defined as follows: S:=\{t\in [0,\infty):nt\...
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 ...
800 views

### Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

I've posted this question on Math Stack Exchange, but I want to bring it here too, because 1) the proof seems missing in the literature, although they are some sporadic mentions and 2) maybe it ...
792 views

### Uniform distribution of points on Riemannian manifolds

Recently, I came across a beautiful paper by Arnol'd and Krylov (Uniform distribution of points on a sphere...) that contains the following theorem: Theorem: Let A and B be two rotations of the ...
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 ...
775 views

### Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?

Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$. Does anyone know of an ...
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 ...
583 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 ...
1k 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 ...
1k 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 ...
664 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 ...
2k 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 ...
828 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 ...
1k 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 ...
441 views

### Entropy of composition

I asked this at math.stackexchange.com, but got no answers. Let $(X,B,\mu)$ be a probability space. Let $T,S:X→X$ be two measurable measure preserving maps that commute (i.e $TS=ST$). Let $A$ be a (...
520 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 $\mathbb{R}^2$ ...
### 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 (...