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

**2**

votes

**2**answers

1k views

### Almost periodic functions in Tao's ergodic proof of Szemerédi's theorem

Do we say that a function $f$ is uniformly almost periodic in the aforementioned proof if $f$ is bounded (in the sense that $||f||_{L^\infty}\leq 1$) and that there exists a natural number $d>0$ ...

**4**

votes

**2**answers

695 views

### distribution of {na} when a is irrational number

(by $\{x\}$ I mean the fraction part of the real number $x$)
If $a$ is an irrational number and $n$ is a integral number, what is the distribution of $\{na\}$? I'm asking for some continuous function ...

**8**

votes

**2**answers

521 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 ...

**4**

votes

**2**answers

665 views

### Poincaré recurrence; Time Return

Hello everybody! Recently I start a reading of a survey by Benoit Saussol,
AN INTRODUCTION TO QUANTITATIVE POINCARE RECURRENCE IN DYNAMICAL SYSTEMS, I am interested in references (Papers) Basics ...

**4**

votes

**2**answers

406 views

### Unique equilibrium states for systems without specification

Let $X$ be a compact metric space and let $f\colon X\to X$ be a continuous expansive map. Let $\mathcal{V}$ denote the space of Hölder continuous potential functions $\phi\colon X\to ...

**6**

votes

**3**answers

701 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 ...

**6**

votes

**3**answers

502 views

### What do singular, atomless invariant measures of $\times d$ look like?

Consider the circle map $\times d:x\mapsto dx \mod 1$. The lebesgue measure is the only absolutely continuous invariant probability measure, but this map has many other invariant measures. Of course, ...

**5**

votes

**3**answers

522 views

### Poincare Recurrence and Dense Sets

This is kind of a spin-off of the question asked here. Take the interval $X:=[0,1]$ with $\mu$ being standard Lebesgue measure. Let $f$ be a measure preserving map $f:[0,1]\rightarrow [0,1]$. The ...

**7**

votes

**2**answers

432 views

### How quickly will billiard trajectories cluster?

Suppose you launch $n$ point-particles on
distinct reflecting nonperiodic billiard trajectories
inside a convex polygon. Assume they all have the same speed.
Define an $\epsilon$-cluster as a ...

**5**

votes

**0**answers

408 views

### Reference request: natural extensions of topological dynamical systems

I am currently writing a paper in which I need to use the following fact: if $T \colon X \to X$ is a uniquely ergodic transformation of a compact metric space, and $\mathcal{A}$ is a continuous ...

**5**

votes

**4**answers

493 views

### orbits in locally compact group

As everyone knows if $x\in S^1$, then the set $\{ x^n \}$ is either finite or dense. Under which condition is true for any other locally compact group, i.e if $G$ is a locally compact group, and $x\in ...

**7**

votes

**3**answers

722 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 ...

**26**

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 ...

**21**

votes

**1**answer

1k 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 ...

**2**

votes

**1**answer

296 views

### Poincare Recurrence Theorem on Infinite Measure Space

Suppose that $(\Omega,\mathcal{A},\mu)$ is a $\sigma$-finite measure space of infinite measure and $T:\Omega\to\Omega$ a measure-preserving transformation with measurable inverse. Let be $\Omega_k\in ...

**1**

vote

**1**answer

181 views

### sum of Perron-Frobenius operators

My operator is the transfer operator $P$ on $L^1$ functions defined on compact $X$. It is the pre-dual of the operator $U:L^∞ \rightarrow L^∞$ defined by $U(ϕ)=ϕ\circ f$, for a fixed map f on X. I ...

**3**

votes

**0**answers

195 views

### links and interactions between different approaches to (super-)rigidity

By super-rigidity I mean some theorems concerning the arithmetic subgroups in semi-simple Lie groups. According to Margulis "Discrete subgroups of semi-simple Lie groups" (the book published by ...

**5**

votes

**4**answers

345 views

### Transitive shifts with multiple fully supported MMEs

This is a sequel to my earlier question, where I asked for an example of a shift space that is mixing but not intrinsically ergodic -- that is, it has multiple measures of maximal entropy (MMEs). ...

**14**

votes

**3**answers

458 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 ...

**7**

votes

**2**answers

413 views

### A non-standard ergodic limit

Suppose $T$ is an ergodic measure-preserving transformation on a measure space $(X,\Sigma,\mu)$, and $f\in L^1(\mu)$. Does the limit
$\lim_{X\to\infty} \pi(X)^{-1}\sum_{p\leq X} f(T^{p}x)$
exist ...

**8**

votes

**5**answers

779 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 ...

**4**

votes

**2**answers

566 views

### Is the distance function from a point to the Mandelbrot set computable?

There is at least one result saying that the Mandelbrot set is undecidable, and there might be more, but I think it (or they all) use real computation rather than Turing machines. This makes some ...

**5**

votes

**0**answers

250 views

### Central extensions of automorphisms of Bruhat-Tits trees

This is the first time I am using Mathoverflow and I am still learning how to use it.
That is why I want to begin with a curious question:
Does the group of automorphisms of a Bruhat-Tits tree have ...

**4**

votes

**4**answers

302 views

### Inflating/Deflating diffeomorphism

I'm interesting in find (if it exists) any manifold with volume form and any diffeomorphism on it such that for any ball B corresponding sequence of B-iterated volumes (just measures of $T^i(B)$, ...

**9**

votes

**3**answers

992 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 ...

**5**

votes

**0**answers

354 views

### Is there a continuous-time version of Kingman's subadditive decomposition theorem?

Kingman's subadditive ergodic theorem (see this article) states that if $x_{m,n}$ is a real valued process indexed on the set of pairs of non-negative integers $m < n$ satisfying:
$x_{l,n} \le ...

**4**

votes

**1**answer

374 views

### Invariant measures for $1$-dimensional discrete dynamical systems

The image below was created using the current release of the
visualization program 3D-XplorMath (available
by clicking here. )
It is an image of the Feigenbaum Tree, on
which is superimposed a ...

**5**

votes

**4**answers

805 views

### Product Measure Only Possible Measure?

Let $X$ be a separable complete metric space and $Z$ be the set of all integers. Let $\nu$ be a Borel probability measure on $X^Z$ invariant under the shift function $S:X^Z \to X^Z$. Is it necessarily ...

**2**

votes

**1**answer

267 views

### exactness of the Gauss transformation

Dear all,
I would like to know if the Gauss transformation T(x) = fractional part of 1/x, x in (0,1) (with the Gauss invariant probability measure) is an exact endomorphism (in the sense of Rokhlin). ...

**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 ...

**9**

votes

**2**answers

349 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 ...

**5**

votes

**1**answer

205 views

### Minimal elements of minimal R^k actions

C. Pugh and M. Shub showed in 1971 that, given an ergodic action of $G=\mathbb{R}^k$ on some separable finite measure space $(X,\mu)$, then all elements of $G$ , off a countable family of hyperplanes, ...

**4**

votes

**1**answer

454 views

### Birkhoff ergodic theorem for dynamical systems driven by a Wiener process

At the risk of asking a stupid question I have the following problem.
Suppose I have a measure preserving dynamical system $(X, \mathcal{F}, \mu, T_s)$, where
$X$ is a set
$\mathcal{F}$ is a ...

**25**

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 ...

**20**

votes

**5**answers

794 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 ...

**5**

votes

**4**answers

476 views

### When is there a natural Riemannian metric whose measure preserves a self-diffeomorphism?

Let $M$ be a compact Riemannian manifold with metric $g$ and let $f \in Diff(M)$.
Under what circumstances is there a natural metric $g_f$ s.t. the associated smooth measure $\nu_f$ is preserved by ...

**7**

votes

**1**answer

292 views

### Ergodic splitting in L_p

I have a curiosity on the Ergodic decomposition given by the von Neumann's theorem:
$$L^2(X,\Sigma,\mu)=L^2(X,\Sigma_T,\mu)\oplus\overline{\{f-f\circ T\ :\ f\in L^2(X,\Sigma,\mu)\}},$$
that occurs ...

**5**

votes

**2**answers

630 views

### topologically mixing subshifts without ergodic measures

Are there examples of subshifts (that is, closed shift-invariant subsets of the full shift {$1...n$}${}^{\mathbb{Z}}$) on which the shift is topologically mixing, which admit a shift-invariant ...

**5**

votes

**2**answers

496 views

### A topologically mixing subshift with multiple measures of maximal entropy

Let Σp={1,...,p}ℤ be the full shift on p symbols, and let X ⊂ Σp be a subshift -- that is, a closed σ-invariant subset, where $\sigma\colon \Sigma_p\to \Sigma_p$ is the ...

**7**

votes

**2**answers

234 views

### a.e. convergence of the powers of an operator built from rotations

Consider two numbers $a,b\in R/Z$ and some integer $p\geq 1$. Let $T:L^p(R/Z)\rightarrow L^p(R/Z)$ be the operator given by
$$T(f)(x)=1/2(f(x+a)+f(x+b))$$
For which values of $a,b$ do we have almost ...

**4**

votes

**2**answers

345 views

### Do there exist Markov partitions with (nearly) uniform SRB measures?

Let $M$ be a compact, finite-dimensional Riemannian manifold, let $T: M \rightarrow M$ be an Anosov diffeomorphism, and let $\mu$ be a Sinai-Ruelle-Bowen (probability) measure. Write $\mathcal{R} = \{ ...

**6**

votes

**4**answers

981 views

### On The Convergence of Ergodic Measures

I would like to know an example (not using the Gibbs measure Theory) of a sequence of measures $\mu_n:\mathcal B\to[0,1]$ , where $\mathcal B$ is the $\sigma$-algebra of the borelians of a compact ...

**3**

votes

**2**answers

456 views

### How Can I Tell when A Subgroup of a Lie Group is Generated by Unipotents?

I'm trying to understand the proof of the Oppenheim conjecture using Ratner's theorem, and I don't immediately see why $SO(2,1)$ is generated by unipotents. Why is $SO(2,1)$ generated by unipotents? ...

**4**

votes

**1**answer

324 views

### Is the average first return time of a partitioned ergodic transformation just the number of elements in the partition?

For some reason my thinking is very fuzzy today, so I apologize for the following rather silly question below...
Let $T$ be an ergodic transformation of $(X,\Omega, \mathbb{P})$ and let $X$ be ...

**5**

votes

**4**answers

475 views

### Fundamental domains of measure preserving actions

Suppose a finite group $G$ acts on a standard probability space $(X, \mu)$ by measure-preserving actions (i.e. $\mu(g(A)) = \mu(A)$ for all $g \in G$ and $A \subset X$ measurable). In addition, ...

**3**

votes

**1**answer

260 views

### Finitarily Markovian Finite Factors of Bernoulli Schemes

By processes, I mean discrete, stationary stochastic processes, that is $(X,\mathcal{U},\mu,T)$ where $X$ is the set of doubly infinite sequences of some alphabet $A$, $\mathcal{U}$ is the ...

**3**

votes

**0**answers

178 views

### What is known about first return times to Markov partitions for Anosov diffeomorphisms?

Consider an Anosov diffeomorphism $T: M \rightarrow M$ and a corresponding Markov partition $\mathcal{R}$ of $M$. For $x \in M$, let $\mathcal{R}(x)$ denote the element of $\mathcal{R}$ containing $x$ ...

**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 ...

**6**

votes

**1**answer

451 views

### A regularity property of transition matrices for the cat map

I've noticed a rather strange phenomenon (not important for my particular research, but interesting) and wouldn't be surprised if someone more versed in symbolic dynamics (i.e., just about anyone who ...

**4**

votes

**4**answers

413 views

### Does it help to learn statistical mechanics in order to learn thermodynamic formalism?

Does it help to learn statistical mechanics or thermodynamics (as in physics or mathematical physics) in order to learn thermodynamic formalism: the study of equilibrium states, Gibbs measure, maximal ...