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

**13**

votes

**0**answers

377 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

465 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

361 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

210 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

223 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

296 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

124 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 ...

**5**

votes

**0**answers

71 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

105 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

173 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

169 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

175 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 ...

**5**

votes

**0**answers

204 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

400 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

439 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

253 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

372 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

74 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 ...

**4**

votes

**0**answers

82 views

### Multifractal Analysis and Dimension Spectrum of Unions

Consider the classical Multifractal Analysis, and the decomposition of the state space $X$ into level sets
$$X=\bigcup_{\alpha}\left\{x\mid d_\mu(x)=\alpha\right\}\cup\left\{x\mid d_\mu(x) ...

**4**

votes

**0**answers

369 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

198 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

31 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

101 views

### Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...

**3**

votes

**0**answers

92 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

92 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

99 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

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

180 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

85 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

97 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

77 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

79 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

98 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 ...

**2**

votes

**0**answers

122 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

88 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 ...

**2**

votes

**0**answers

167 views

### Reference request: stationary measures as convex combinations of ergodic measures

Does anyone know a good reference for the fact that a stationary probability measure is a convex combination of the stationary and ergodic probability measures?
I have found some references for the ...

**2**

votes

**0**answers

76 views

### Ergodic actions with co-finite stabilizers

Let $G$ be a locally compact, second countable group acting on a standard probability space $(X,\nu)$, and let $\nu$ be $G$-invariant. Let $G_x = \{g \in G\,:\, gx=x\}$ denote the stabilizer of $x \in ...

**2**

votes

**0**answers

141 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 ...

**2**

votes

**0**answers

155 views

### Partitions of central sets via dynamical systems

In the book "Recurrence in Ergodic Theory and Combinatorial Number Theory", 1981, Furstenberg introduced the notion of central sets.
He proved in Theorem 8.8 that in each finite partition of ...

**2**

votes

**0**answers

178 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 ...

**2**

votes

**0**answers

76 views

### adic periodic approximation of a dynamical system

Let $P$ be the Pascal adic transformation. The cutting and stacking construction of $P$ corresponds to a ``Pascal periodic approximation'' of $P$: a sequence $(P_n)$ of periodic automorphisms strongly ...

**2**

votes

**0**answers

401 views

### Virtual nilpotent ergodic average

In a recent paper of Miguel N. Walsh,"Norm convergence of nilpotent ergodic averages"(http://arxiv.org/abs/1109.2922v2),the author gives a proof of the fact that multiple polynomial ergodic averages ...

**1**

vote

**0**answers

58 views

### Markov chain with state space $\mathbb{R}^+$

I am studying Markov chains on general state spaces (Markov Chains and Stochastic Stability by Meyn and Tweedie), but there are a few things that are not yet completely clear to me.
Suppose I have a ...

**1**

vote

**0**answers

100 views

### Name/terminology for a relationship between group actions

Let $G$ and $H$ be groups, both acting on a set $X$. Suppose that there is a homomorphism $\phi:G\to H$ such that for every $g\in G$ and $x\in X$, $g\cdot x = \phi(g)\cdot x$. Is there a name for this ...

**1**

vote

**0**answers

79 views

### A argument related measurable partitions in dynamic system

$X$ is a compact metric space, and $T:X\rightarrow X$ be a continuous map, which is finite to one. Denoted by$ X_{0}$ the set of all points $x\in X$, such that for all sufficiently small ...

**1**

vote

**0**answers

136 views

### Entropy of factors of Bernoulli schemes

Let $X$ be a Bernoulli scheme. A factor $\psi: X \to Y$ is finitary if for almost every $x \in X$ there exist integers $m \leq n$ such that the zero coordinates of $\psi(x)$ and $\psi(x')$ agree for ...

**1**

vote

**0**answers

199 views

### On the set of infinite measures

My question is about the structure of the set of infinite Borel measures on compact metric spaces invariant with respect to a homeomorphism.
Let $T$ be a homeomorphism of a compact metric space $X$ ...

**1**

vote

**0**answers

155 views

### Entropy of Bernoulli walks on semi-groups.

Consider the Fibonacci semi-group $<L,R|LRR=RLL>$ with a Bernoulli walk $P(R)=p, P(L)=1-p$. Is the entropy $H(p)$ an unimodal function with maximum at p=0.5? Is this true for all finitely ...

**1**

vote

**0**answers

222 views

### weak mixing and spectral theorem

I'm trying to prove an equivalent statement about weak mixing transformations that relies on the spectral theorem, but I can't find a reference to fill in the last details. A hint for solving it or ...

**1**

vote

**0**answers

170 views

### Ergodicity of non-homogeneous “rotations”

It is well known that a rotation $f(z)=e^{i\theta}z$ of the unit circle, is ergodic if and only if $e^{i\theta}$ is not a root of unity.
Now, what happens if we let $\theta$ depend on $z$ (say, ...