Questions tagged [ergodic-theory]

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

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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 ...
Ngoc Mai Tran's user avatar
27 votes
0 answers
807 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 ...
Joseph O'Rourke's user avatar
18 votes
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472 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 ...
Joseph O'Rourke's user avatar
12 votes
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279 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 ...
dhy's user avatar
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12 votes
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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 ...
Dominik Kwietniak's user avatar
11 votes
0 answers
189 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: ...
Sophie M's user avatar
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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 ...
Fan's user avatar
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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 $r$,...
alvarezpaiva's user avatar
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10 votes
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968 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 ...
Steven Deprez's user avatar
9 votes
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184 views

For measure-preserving systems, is countable generatability of the invariant $\sigma$-algebra equivalent to almost all points being periodic?

Let $X$ be a second countable Hausdorff topological space, let $T \colon X \to X$ be a Borel-measurable map, define the $\sigma$-algebra $\mathcal{I}=\{A \in \mathcal{B}(X) : T^{-1}(A)=A\}$, and for ...
Julian Newman's user avatar
7 votes
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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 $\alpha$, ...
Dominik Kwietniak's user avatar
7 votes
0 answers
305 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 ...
Stéphane Laurent's user avatar
7 votes
0 answers
699 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_{...
Pablo Lessa's user avatar
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6 votes
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Countable companions for Polish locally compact groups and their orbit equivalence relations

In "Countable sections for locally compact group actions" (Ergod. Th. & Dynam. Sys., 1992), Kechris proved that if $G$ is a Polish locally compact group acting in a Borel way on a ...
Iian Smythe's user avatar
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Direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic

I wonder if there are any direct proof that $g_t=\operatorname{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic (or even stronger,...
No One's user avatar
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Construction of minimal zero entropy measure-theoretically strong mixing subshift?

Does anyone know of a construction of a subshift (over $\mathbb{Z}$) which is (1) minimal (2) zero (topological) entropy (3) measure-theoretically strong mixing (for some measure)? I am in particular ...
Ronnie Pavlov's user avatar
6 votes
0 answers
254 views

A density result for arithmetic progressions

Note: By upper/lower density, we shall mean the upper/lower asymptotic density as given here. Question: For any subset $S \subset \mathbb N$ with positive upper density, does there exists a $\...
Nate River's user avatar
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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(\...
Dominik Kwietniak's user avatar
6 votes
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145 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 ...
Aleksei Kulikov's user avatar
6 votes
0 answers
342 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 $f^{-1}(x)$, ...
Adam's user avatar
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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 ...
Maurizio Moreschi's user avatar
6 votes
0 answers
606 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 ...
coudy's user avatar
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236 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 ...
Dominik Kwietniak's user avatar
6 votes
0 answers
239 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 ...
Vladimir's user avatar
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6 votes
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383 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 ...
Yellow Pig's user avatar
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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 ...
Alin Galatan's user avatar
6 votes
0 answers
320 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 $\mu$...
Luka Thaler's user avatar
6 votes
0 answers
379 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 ...
FelipeG's user avatar
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0 answers
296 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 $\...
Tom LaGatta's user avatar
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6 votes
0 answers
351 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 $\...
Łukasz Grabowski's user avatar
6 votes
0 answers
1k 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 ...
Ian Morris's user avatar
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6 votes
0 answers
285 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 ...
Hadi's user avatar
  • 731
6 votes
1 answer
276 views

Sequence of digits of powers of two

Elementary number theory tells us a lot about the final digits of the powers of two, and ergodic theory (more specifically the theory of equidistribution of points in the orbit of an irrational ...
James Propp's user avatar
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5 votes
0 answers
164 views

Counterexamples to the Ahlfors measure conjecture in higher dimensions

Let $\Gamma<SO(3,1)$ be a finitely generated, discrete group of isometries of $\mathbb H^3$. By work of Agol, Calegari, Canary, and Gabai, the limit set of $\Gamma$ is either the entire sphere $S^2\...
Yankl's user avatar
  • 317
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0 answers
86 views

Lower bound for nonconventional ergodic averages in finite fields

Let $p$ be a sufficiently large prime number and $f\colon\mathbb{F}_{p}\to\mathbb{R}_{\geq 0}$ be a function bounded by 1 such that the average of $f$ over the finite field $\mathbb{F}_{p}$ is at ...
Wenbo Sun's user avatar
5 votes
1 answer
376 views

Positiveness of the largest Lyapunov exponent

Let $\alpha\in \mathbb{R} / \mathbb{Q}$, let $A(x)$ be the $2$-by-$2$ matrix $$ A(x)=\begin{pmatrix} \dfrac{1}{{\lambda}^2}-2 \cos 2\pi x -1& 2\lambda \cos 2\pi x-\dfrac{1}{{\lambda}} \\ \dfrac{...
xia  xu's user avatar
  • 131
5 votes
0 answers
148 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 ...
Ilya Gekhtman's user avatar
5 votes
0 answers
154 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 ...
Joseph O'Rourke's user avatar
5 votes
0 answers
171 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 ...
jason's user avatar
  • 553
5 votes
0 answers
829 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 ...
Ryan Alweiss's user avatar
5 votes
0 answers
138 views

Law of Large Numbers for the Tasep from a Bernoulli Configuration (Rost's Theorem)

Let $(\eta_{t}^{\rho})_{t\geq 0}$ be a totally asymmetric simple exclusion process (TASEP) from an initial configuration distributed according to the Bernoulli measure $\nu_{\rho}$ on $\{0,1\}^{\...
Nahuel Albarracín's user avatar
5 votes
0 answers
80 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 ...
user avatar
5 votes
0 answers
240 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 ...
David Handelman's user avatar
5 votes
0 answers
255 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 $\mathbb{H^2/\Gamma}$ ...
A. S.'s user avatar
  • 518
5 votes
0 answers
419 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. ...
Florin Radulescu's user avatar
5 votes
0 answers
194 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 $\mu_{n}(U)=\max_{t\in S^{1}}\frac{1}{n}\cdot|\{k\in\{1,...,n\}|t\cdot e^{\frac{2\pi ik}{n}}\in U\}|$. ...
Joseph Van Name's user avatar
5 votes
0 answers
235 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 $1<p<\infty$, ...
Alan's user avatar
  • 1,514
5 votes
0 answers
141 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 ...
Tom Kempton's user avatar
5 votes
0 answers
223 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 ...
Tom Kempton's user avatar
5 votes
0 answers
201 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 ...
A Blumenthal's user avatar

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