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

**14**

votes

**0**answers

427 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 ...

**10**

votes

**0**answers

497 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

**0**answers

382 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 ...

**7**

votes

**0**answers

230 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 ...

**6**

votes

**0**answers

194 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

**0**answers

155 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 ...

**6**

votes

**0**answers

244 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**

votes

**0**answers

191 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 ...

**6**

votes

**0**answers

302 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 ...

**5**

votes

**0**answers

232 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. ...

**5**

votes

**0**answers

143 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**

votes

**0**answers

170 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 $1<p<\infty$, ...

**5**

votes

**0**answers

113 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 ...

**5**

votes

**0**answers

75 views

### Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic?
Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...

**5**

votes

**0**answers

129 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**

votes

**0**answers

183 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 ...

**5**

votes

**0**answers

186 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**

votes

**0**answers

206 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 ...

**5**

votes

**0**answers

450 views

### Density of strictly ergodic measures in the d-bar topology

I am currently studying a problem which deals with cocycles of highly
noncompact operators on Hilbert space, with the base transformation
being the full shift on two symbols. In my particular ...

**5**

votes

**0**answers

519 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

**0**answers

257 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 ...

**5**

votes

**0**answers

397 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

**0**answers

229 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 ...

**4**

votes

**0**answers

373 views

### entropy preserving finitary factor maps of Bernoulli schemes

Let $X=\{0,1\}^\mathbb{Z}$ with measure $\mu=(p,1-p)^{\mathbb{Z}}$.
Let $(\phi(x))_i=(x_i+x_{i+1})$mod$2$.
If $p=1/2$, then $\phi(X)=X$. If $p \not = 1/2$, then $\phi(X)$ is not a Bernoulli scheme ...

**4**

votes

**0**answers

200 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 ...

**3**

votes

**0**answers

125 views

### On the decay of correlations of an ergodic sequence over the set $X_{0}=0$

The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It ...

**3**

votes

**0**answers

76 views

### Topological pressure for subshifts on a countable alphabet

Apologies for asking two similar questions within a week of each other, I had hoped that asking a finite alphabet version of this question would lead to enlightenment but unfortunately it didn't.
...

**3**

votes

**0**answers

156 views

### Almost periodic sequence

Say that a real sequence ${(x_k)}_{k \in \mathbb{Z}}$ is almost periodic if the set of all its shifted sequences ${\left\{{(x_k)}_{k+n \in \mathbb{Z}}\right\}}_{n \in \mathbb{Z}}$ is relatively ...

**3**

votes

**0**answers

62 views

### Ferenczi: minimal, uniquely ergodic, sublinear complexity systems are not strongly mixing

The following result is on page 26 of this paper by Ferenczi [PDF].
Corollary 3. A minimal and uniquely ergodic system of sub-affine complexity cannot be strongly mixing (i.e., $\mu(T^nA \cap B) ...

**3**

votes

**0**answers

105 views

### Equidistribution of double coset

Let $G=PGL_n(\mathbb{R})$, $K=PO_n(\mathbb{R})$ and $X=G/K$. Also suppose $\Gamma=SL_n(\mathbb{Z})$ acts on the left of $X$. We define a typical Hecke operator on $L^2(\Gamma\backslash X)$ by the ...

**3**

votes

**0**answers

100 views

### Any references on infinite-dimensional Fourier-Plancherel theory?

Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...

**3**

votes

**0**answers

114 views

### Pointwise ergodic theorem for amenable semigroups

Using tempered Folner sequences one may show a pointwise ergodic theorem for amenable groups.
(see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full)
Is there a similar ...

**3**

votes

**0**answers

84 views

### Best convergence rate for convolutions on $\mathbb{Z}_p$

Suppose, that we have sequence of i.i.d variables $X_1,\ldots,X_n$ taking values in $\mathbb{Z}_p$ such that $d_{TV}(X_1,U) < \delta$.
How fast, in terms of $\delta$ and $n$ does the sum ...

**3**

votes

**0**answers

166 views

### Markov operators and existence of ergodic measures

My question refers to the yesterday's question (see here)
of John Learner and goes as follows:
Can we deduce the existence of an ergodic measure if we know that an invariant measure exists, but the ...

**3**

votes

**0**answers

111 views

### Are irrational multiples of central sets again central?

Let me begin by giving the relevant definitions. A set $A \subset \mathbb{N}$ is said to be central if and only if there exists a topological system $(X,T)$ (with $X$ a compact metric space, $T$ a ...

**3**

votes

**0**answers

190 views

### The ergodic theorem and Lorentz norms

Let $\Omega$ be a probability space, and $\{ \tau \}_{y\in \mathbb R^d}$ an ergodic group of measure-preserving transformations, $\tau_y:\Omega \to \Omega$.
The ergodic theorem says that if $f \in ...

**3**

votes

**0**answers

250 views

### A general Lipschtiz potential can be specified by a Gibbs specification ?

I want to consider one-dimensional system on the lattice $\mathbb{L}=\mathbb{N}$.
Let be $A:(\mathbb{S}^1)^{\mathbb{L}}\to\mathbb{R}$ a lipschtiz potential. Consider the Ruelle operator
$$
...

**3**

votes

**0**answers

186 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 $x$ ...

**2**

votes

**0**answers

86 views

### Equivalence relations that are both not treeable and amenable

Hyperfinite equivalence relations are treeable. For the case of uncountable relations, I was wondering if there is a reference to (or simple proof of) the following statement: Let $E$ be a (possibly ...

**2**

votes

**0**answers

69 views

### Non-ergodic Dye Theorem for orbit equivalent automorphisms

The Dye Theorem states that any two free ergodic p.m.p automorphisms of a standard probability space are orbit-equivalent.
Question: Is there a version of the above theorem for non-ergodic ...

**2**

votes

**0**answers

82 views

### Do the Birkhoff averages of a measurable stationary homogeneous Markov process in continuous time “converge to the right limit”?

[I've decided to rewrite the question, to make the essential point clearer.]
Let $\,\mathbb{R}^{[0,\infty)}:=\{(x_t)_{t \geq 0} : x_t \in \mathbb{R} \ \, \forall t\}$. We say that a set $Y \subset ...

**2**

votes

**0**answers

136 views

### Discontinuity of Radon-Nikodym derivative for Patterson-Sullivan measures for word metrics on Gromov hyperbolic groups

Let $\Gamma$ be a Gromov hyperbolic group coming endowed with a word metric coming from some finite generating set. Let $\nu$ be an associated Patterson-Sullivan measure (quasi-conformal density).
I ...

**2**

votes

**0**answers

113 views

### A question related to metric Diophantine approximation

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that
$$
\left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q}
$$
has ...

**2**

votes

**0**answers

98 views

### Under what conditions do time averages of ergodic transformations satisfy a central limit theorem?

Let $(X, \mu)$ be a probability space and $T:X\rightarrow X $ an ergodic transformation, i.e. $T$ is measure preserving and the only $T$ invariant subspaces have either measure $0$ or measure $1$ ...

**2**

votes

**0**answers

98 views

### Pointwise convergence of ergodic averages of unconventional conditional expectations

Let $(X_i,Y_i)_{i\in\mathbb{Z}}$ be a finite-valued stationary process whose $\sigma$-algebra of tail events is trivial. Let $\mathcal{F}_n^m$ be the $\sigma$-algebra generated by $X_n,\dots,X_m$ ...

**2**

votes

**0**answers

102 views

### Is $\text{Bow}(X,T)$ a Banach Space?

Let $X=\{0,1\}^{\mathbb{N}}$ be the sequence space and $T:X\to X$ the left shift mapping. Define the vector space $\text{Bow}(X,T)$ as
$$
\text{Bow}(X,T)=\{f\in C^{0}(X);~\sup_{n\in ...

**2**

votes

**0**answers

79 views

### Null sets visited infinitely often by trajectories of the shift dynamical system

Let $(G,\circ)$ be a Polish group, with identity $e$.
Let $\Omega$ be the set of continuous functions $\omega:\mathbb{R} \to G$ such that $\omega(0)=e$.
For each $t \in \mathbb{R}$, define the ...

**2**

votes

**0**answers

92 views

### uniquely ergodic hyperbolic invariant set

The question is to classify uniformly hyperbolic invariant sets supporting uniquely ergodic invariant measure. The only examples that I expect are: fixed points, periodic orbits and Cantori(Denjoy ...

**2**

votes

**0**answers

135 views

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

Let $T : X \to X$ 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 ...

**2**

votes

**0**answers

108 views

### order of convergence of the conditional entropy (3)

I'm sorry for having open two questions which have been solved by elementary counter-examples provided by @AnthonyQuas. Actually I'm not an expert in information theory and I expected that a positive ...