# Questions tagged [ergodic-theory]

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

226
questions with no upvoted or accepted answers

**46**

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### 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 ...

**26**

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713 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**

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**0**answers

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 ...

**12**

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**0**answers

256 views

### Statistics for rational points on curves of genus $g$ over $\mathbb{F}_q$, $g\gg q$

Consider the distribution of the number of $\mathbb{F}_q$ points as I range over smooth projective curves of genus $g$ (defined over $\mathbb{F}_q$). If $q\gg g,$ the Hasse-Weil bounds give me a lot ...

**11**

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1k views

### Weak$^*$ convergence of measures vs. convergence of supports

Let $X$ be a compact metric space and let $\mathcal M(X)$ denote the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\text{supp} \mu$ for the support of $\mu$. It is easy to ...

**10**

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503 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**

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874 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**

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98 views

### Factor map between subshifts preserving topological pressure (or measure-theoretic entropy)

Let $G$ be a countable amenable group and let $X,Y$ be subshifts with finite alphabet over $G$. Suppose that $h(X) = h(Y)$ (equal topological entropy). I am interested in continuous factor maps $\pi: ...

**9**

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241 views

### Mackey's Program on Algebraic Ergodic Theory

I knew about Mackey's Program from Arnold's book Random Dynamical Systems and it referred to K. Schmidt's book Algebraic Ideas in Ergodic Theory, which was published in 1990. However, that is the ...

**7**

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150 views

### Possible Birkhoff spectra for irrational rotations

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, ...

**7**

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295 views

### Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action

Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...

**6**

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111 views

### Countable-to-one factors of measure preserving systems do not change entropy

It is known that if $\psi$ is a factor map between probability measure preserving systems $(X,\mathscr{X},\mu,T)$ and $(Y,\mathscr{Y},\nu,S)$ is countable-to-one almost everywhere, then $h(\mu,T)=h(\...

**6**

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117 views

### Quantitive and computational improvement of the Oseledets multiplicative ergodic theorem for irrational rotation

Consider irrational rotation $T:S^1\to S^1, T(x) = x + \alpha$ where $\alpha\notin \mathbb{Q}$ (you may assume additional number theoretic properties of $\alpha$, say $\alpha = \sqrt{2}$ is already ...

**6**

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252 views

### $C^{1+\epsilon}$ conjugacy of expanding map on circle

A continuously differentiable map $f:S^{1}\rightarrow S^{1}$ is called expanding if $|f^{'}(x)|>1$ for all $x\in S^{1}$.
We can define the degree of f, def(f) to be number of preimage $f^{-1}(x)$, ...

**6**

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188 views

### Counting lattice points in adelic spaces

Let $\mathbb{A}$ denote the ring of adeles of $\mathbb{Q}$, let $\mu$ be the Haar measure of $\mathbb{A}$, and let $\|\cdot\|_{\infty}$ denote the sup-norm of the components in the Archimedean ...

**6**

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454 views

### Ergodicity and dense orbits

Consider a compact separable Hausdorff space $X$ endowed with a finite
Radon measure $\mu$ of full support and a continuous measure-preserving
ergodic transformation $T$. Is there a dense orbit for ...

**6**

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229 views

### Cartesian square root of a measure preserving action

Let $G \curvearrowright (X,\nu)$ be probability measure preserving action of a countable discrete group. When does there exist a probability measure preserving action $G \curvearrowright (Y,\mu)$ such ...

**6**

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376 views

### When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...

**6**

votes

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409 views

### Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep

Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...

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308 views

### Measure theoretic entropy

I don't know if this is an elementary question or not. In what follows all maps are continuous
Suppose that $P:\mathbb{C}\rightarrow\mathbb{C}$ is a complex polynomial of degree $d>1$ and let $\mu$...

**6**

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351 views

### Do ergodic isometries have discrete spectrum?

Let $X$ be a metric space, $\mu$ a Borel probability measure, and
$T:X\rightarrow X$ be an ergodic measure preserving isometry.
Is $(X,\mu,T)$ measure theoretically isomorphic to a minimal isometry ...

**6**

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289 views

### Generating stationary, ergodic random fields on a homogeneous space

Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel $\...

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345 views

### "topological" conjugacy of group automorphisms

In the paper "Orbit Equivalence and Topological Conjugacy of Affine
Actions on Compact Abelian Groups", S. Bhattacharya shows (Theorem 3) the following:
Theorem. Given two actions $\alpha$ and $\...

**6**

votes

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892 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 ...

**6**

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277 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 ...

**6**

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618 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 x_{...

**5**

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103 views

### Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?

It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure.
Marc Peigné [1] constructed examples of geometrically infinite hyperbolic ...

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142 views

### Are topologically free and essentially free equivalent for minimal spaces with invariant measures?

Suppose $G$ is a discrete group acting by homeomorphisms on a compact Hausdorff space $X$, such that the action is minimal. Fix an invariant Radon measure $\nu$ on $X$. Is topologically free (the ...

**5**

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148 views

### Central limit theorem versus entropy in dynamical systems context

A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem.
Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...

**5**

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196 views

### Looking for a counterexample for Ruelle's inequality on compact manifold

Let $M$ be a compact differentiable manifold, and $f:M\to M$ be a $C^1$- smooth diffeomorphism.
If Assume that $\mu$ be a $f$-invariant probability measure on $M$.
Then D.Ruelle proved that
$$
h_\...

**5**

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775 views

### Increasing sequences in polynomial progressions modulo p

In a random permutation on $n$ elements one expects the largest increasing and decreasing sequences to have size $(2+o(1))\sqrt{n}$. Is it known if this same property holds in sequences given by ...

**5**

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178 views

### Completeness of the space of measures under $d$-bar metric

Does anybody know the reference to a proof of the following fact (which is not hard to prove, but seems to be well-known, see here): The space of shift-invariant measures under Ornstein's d-bar metric ...

**5**

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78 views

### What statistical data/quantities are known about the time spent by a generic orbit of an ergodic system in a fixed set?

By the ergodic theorem, we know that for almost every point, the average time spent by an orbit in a set is equal to the relative measure of that set.
What other information about that time can we ...

**5**

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230 views

### Paths in Pascal's triangle; or balanced $0-1$ initial segments

Here is a problem arising (via a tortuous path) from trying to determine the spectrum of Vershik's adic map on Pascal's triangle (a moderately well-known question: is the spectrum trivial, that is, is ...

**5**

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234 views

### Equidistribution of spheres in $\mathbb{R^2}/\mathbb{Z^2}$

Let $\mathbb{H^2}$ be the hyperbolic upper half place, and let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$ acting on $\mathbb{H^2}$. A proof of the equidistribution of spheres on $\mathbb{H^2/\Gamma}$ ...

**5**

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375 views

### Koopman representation, weakly compact action, Ozawa Popa

Given a weakly compact action (Ozawa-Popa) of a discrete group $\Gamma$ on p.m space $X$, consider the Koopman representation $\pi$ on $L^2(X)$. Compose this representation with the Calkin projection. ...

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186 views

### Characterizations of an exotic measure on the open sets in the circle $S^{1}$

Suppose that $U\subseteq S^{1}$ is open where $S^{1}=\{z\in\mathbb{Z}:|z|=1\}$. Then define $\mu_{n}(U)=\max_{t\in S^{1}}\frac{1}{n}\cdot|\{k\in\{1,...,n\}|t\cdot e^{\frac{2\pi ik}{n}}\in U\}|$. ...

**5**

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228 views

### Is Akcoglu's theorem for power bounded positive operators still an open problem?

I am reading Ulrich Krengel's book, Ergodic Theorems; the theorem of Akcoglu's he mentions of is on page 189, theorem 2.5.
" If $T$ is a positive contraction in a space $L_p$ with $1<p<\infty$, ...

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133 views

### Growth in families of trees

I'm hoping that the question below is simple thermodynamic formalism, but I can't quite make it work. Any help would be very welcome.
Let $\Sigma:=\{0,1\}^{\mathbb N}$ and let $\Sigma^*$ be the set ...

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211 views

### Using topological pressure to determine a subshift of finite type

I am interested in recognising graphs (or matrices, or subshifts of finite type) using topological pressure. Suppose that we play the following game:
${\bf Step 1:}$ I write down an irreducible n x n ...

**5**

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191 views

### Reference for and Properties of the $\alpha$-entropy

Let $T \colon X \to X$ be a continuous map on, say, a compact metric space $X$. Let $\mu$ be an invariant borel measure. Under suitable conditions, a result of Brin and Katok states that $\mu$-almost ...

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212 views

### Quasicompactness of transfer operators associated to IID matrix products

Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...

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965 views

### Fibre Mixing for Dynamical Systems

Hi all,
I'm interested in understanding a fairly difficult theorem of Lindenstrauss Peres and Schlag. In that paper the authors prove that certain dynamical systems related to beta expansions and ...

**5**

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238 views

### Can invariant means be really considered as the generalization of the uniform measure?

I am writing a paper for game theorists where I use (countable) amenable groups to do some things. So I am writing up a preliminary section about countable amenable groups whose main purpose is to ...

**4**

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56 views

### Correspondence between Hoelder cocycles and Hoelder potential functions for noncompact negatively curved manifolds

Let $\tilde{M}$ be the universal cover of a pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary.
When $M$...

**4**

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73 views

### Random walks on the Poincaré disk

Let $G$ be the group of isometries of the Poincaré disk. Let $\mu$ be a probability measure on $G$, and consider $g_1,..,g_n$ i.i.d. random variables on $G$ distributed according to $\mu$. For $z\in \...

**4**

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106 views

### Pocket billiards with balls in general position

There were at least two earlier MO questions about ideal pocket billiards.
(Ideal: frictionless, perfectly elastic collisions.)
Perfectly centered break of a perfectly aligned pool ball rack.
Does ...

**4**

votes

**0**answers

81 views

### Counting simple closed curves

I'm currently trying to understand how to count simple closed curves. I've been reading Alex Wright's survey (https://arxiv.org/pdf/1905.01753.pdf). However, I don't feel like I'm getting the big ...

**4**

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171 views

### QI but not ME : not finitely presented groups!

I would like to know the examples of two groups which are not finitely presented and are quasi-isometric (QI), but they are not measured equivalent (ME) (in the sense of Gromov).
In the literature, ...

**4**

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97 views

### Hecke equidistribution for $L^1$ functions

Put shortly, the question is, does the Hecke equidistribution hold for $L^1$ functions?
To be specific, the version of H.E. I need is
$T_N f \rightarrow \int f d\mu$ as $N \rightarrow \infty$,
...