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

**5**

votes

**0**answers

153 views

+50

### 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 ...

**4**

votes

**3**answers

209 views

### Ergodic theory: from Dynamics to Gibbs measure

I'm trying to understand the ergodic theory approach to statistical mechanics, namely how ergodic measure preserving dynamics lead to the Gibbs measure.
I have a compact space $X$, a probability ...

**0**

votes

**0**answers

144 views

### On a certain set of probability measures on a shift

Denote by $\mathbb{Z}_2=\{0,1\}$ the integers modulo 2.
Let $S:\mathbb{Z}_{2}^{\mathbb{N}}\times\mathbb{Z}_{2}^{\mathbb{N}} \rightarrow \mathbb{Z}_{2}^{\mathbb{N}}$ be the sum $S(a,b) = a+b$, where ...

**2**

votes

**2**answers

82 views

### “Dynamical” spectral gap for the orignal system out of the spectral gap for the induced system

I would like to prove presence of a spectral gap for the transfer (Ruelle-Perron-Frobenius) operator for some non-uniformly hyperbolic dynamical system on the unit interval. Suppose that I know how to ...

**0**

votes

**1**answer

97 views

### entropy growth of invariant measures - General question

In general, given a sequence of shift-invariant measures $\eta_{n}$ on $\{0,1\}^{\mathbb{N}}$ what to do to guarantee this convergence of entropies: $$h(\eta_{n}) \rightarrow \log2?$$
Because I'm ...

**1**

vote

**1**answer

85 views

### entropy and d-bar: how do we estimate continuity?

Let $G = \{0,1\}^{\mathbb{N}} = \mathbb{Z}_{2}^{\mathbb{N}}$ be the Bernoulli space of two symbols, let $\sigma$ be the shift map and $M(G)$ the set of $\sigma$-invariant probabilities. Let $\bar{d}$ ...

**0**

votes

**0**answers

71 views

### when the composition of two ergodic maps is ergodic?

I would like to know if there are sufficient criteria for the composition of two ergodic maps to be still ergodic.
My context is piecewise affine transformations of the torus in arbitrary dimensions

**18**

votes

**2**answers

558 views

### Distribution of the Error term in GH Hardy's “curious result” $\sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$

In an early paper, GH Hardy talks about the distribution of "curious" sum:
$$ \sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$$
where $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...

**4**

votes

**0**answers

69 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 ...

**3**

votes

**2**answers

220 views

### Entropy equals zero?

Imagine you have a shift invariant ($\sigma$-invariant) probability measure $\eta$
in the Bernoulli space $\{0,1\}^{\mathbb{N}}$. Define
$\mathcal{P} = \{[0],[1]\}$;
$\mathcal{P}^{n} = ...

**5**

votes

**1**answer

132 views

### Estimating entropy conditional to an event

Take for example the measure $\mu(n)=n^2$ on $\{1, \ldots, N\}$ and a random variable $X$ distributed according to the probability obtained by normalizing $\mu$.
Does there exists a constant ...

**2**

votes

**1**answer

94 views

### Rate of convergence of ergodic averages related to irrational rotation

Let $\alpha$ be an irrational number, consider the basic dynamical system $T^{n}(0) = \{n \alpha\}$ where $\{.\}$ denotes the fractional part.
Let $a < b$ be two numbers in $[0, 1]$. Then by ...

**5**

votes

**1**answer

200 views

### Importance of Ornstein's isomorphism theorem

"Perhaps the most important parts of the Ornstein theory are criteria for determining whether or not a shift or flow is Bernoulli (a Bernoulli shift, $B_{ct}$ , or $B_{t}^{\infty}$) because it allows ...

**1**

vote

**0**answers

51 views

### Entropy of the Scenery factor in the $T,T^{-1}$ transformation (RWRS)

The $T,T^{-1}$ transformation is an example of a $K$ automorphism which is not Bernoulli (not isomorphic to a shift of an I.I.D. sequence).
Hoffman in ...

**4**

votes

**1**answer

192 views

### The converse of von Neumann's mean ergodic theorem

Recall that the Hilbert space version of von Neumann's mean ergodic theorem says the following.
Let $\{F_n\}_{n=1}^\infty$ be a right Følner sequence of a countable discrete amenable group $\Gamma$ ...

**2**

votes

**1**answer

70 views

### Is there a way to find $\limsup$ and $\liminf$ for ergodic processes almost surely?

Let $\{X_k\}$ be an ergodic process. I know that if $f$ is a smooth real valued function then by Birkoff's ergodic theorem, $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n f(X_k)=\mathbb{E}(f(X_1))\ ...

**0**

votes

**1**answer

101 views

### A regular variation in infinite ergodic theory

I would like to know if there is a notion or an example related to the following situation: a transform $T$ on a space $E$, which is equipped with an infinite measure $\mu$, satisfies $\mu(A\cap ...

**0**

votes

**1**answer

113 views

### joining or coupling

given two shift invariant measures in the Bernoulli space $\{0,1\}^{\mathbb{N}}$, is there a way to construct joinings of them? It's very diffcult, in general, to find exactly the minimal joining i.e, ...

**2**

votes

**1**answer

68 views

### Rajchman measures via strong mixing systems

A Rajchman measure on the unit circle $\mathbb{T}$ is a Borel probability measure $\mu$ with $\lim_{n\to\infty}\hat{\mu}(n)=0$. Where $\hat{\mu}(n)=\mu(z^n)$ for $n\in\mathbb{Z}$ are Fourier ...

**4**

votes

**1**answer

111 views

### When is the time one map of a suspension flow ergodic?

I'm sure the answer to the following question is well known but I couldn't find the answer I needed.
Let $(\Sigma,\sigma)$ denote the full shift on $k$ symbols and let $\mu$ be an invariant measure ...

**0**

votes

**2**answers

108 views

### Ergodic automorphisms of a compact metric abelian group are Bernoulli

In the literature, such as in this article, it is proved that every ergodic automorphism of a compact metric abelian group is Bernoulli. A rotation is not isomorphic to a Bernoulli shift because it ...

**1**

vote

**2**answers

116 views

### Do ergodicity, minimality and equicontinuity on a compact space imply total ergodicity?

Is it true than an aperiodic, ergodic, minimal and equicontinuous dynamical system on a compact metric space is totally ergodic ?
According to some results I found in some books, a rotation on a ...

**1**

vote

**0**answers

45 views

### Small open sets around a point intersecting pieces of orbits

Let $T$ be an ergodic rotation on a compact Abelian group. Can one always find a point $x_0$ and a decreasing sequence of open sets $O_n \searrow \{x_0\}$ such that for every $n$ there exists $K \geq ...

**7**

votes

**2**answers

199 views

### Random suborbits of a rotation

Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...

**2**

votes

**0**answers

59 views

### Decomposition of the space according to the Ergodic Theorem

Given a space $(X, T)$, it is well known that for every $T$-invariant ergodic measure $\mu$, there exists a set $E_\mu$ of $\mu$-measure $1$ s.t. for every "nice" function $f$
$$ ...

**2**

votes

**0**answers

134 views

### binomial coefficients and irrationals

The following, probably either currently impossible to deal with, or
having a negative solution, arose from an ergodic theory question,
presumably itself currently intractible. I am not a number ...

**1**

vote

**2**answers

152 views

### Can I use Birkhoff's Ergodic Theorem for Vector Valued Process?

I have a stationary process $\{u_n\}$ and I have a function $f:\mathbb{R}^L\to \mathbb{R}^+$. I want to evaluate the following limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...

**0**

votes

**0**answers

78 views

### Entropy, convergence and invariant measures

Could you give conditions that a sequence of shift invariant measures $\eta_{n}$ has to satisfy in order to happen this convergence in terms of entropies $h(\eta_{n})$:
$h(\eta_{n}) \rightarrow ...

**1**

vote

**0**answers

53 views

### Ergodic Markov Operator

Given a $\sigma$-additive measure space $(E,\Sigma,\mu)$.
A Markov operator $P : L^1(\mu) \to L^1(\mu)$ is a linear operator with
$ f \geq 0 \Rightarrow Pf \geq 0 $
$ f \geq 0 \Rightarrow ||Pf|| = ...

**1**

vote

**1**answer

120 views

### Entropy, Convergence

imagine you have a sequence $\eta_{n}$ of (shift) invariant measures in the Bernoulli space $\{0,1\}^{\mathbb{N}}$ that satisfy the following: there are a $0<\delta <1$ and an $N$ such that $$n ...

**5**

votes

**0**answers

193 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 ...

**0**

votes

**3**answers

104 views

### Invariant and periodic measures of the random dynamical system on the circle generated by $d\theta_t=dW_t$

Here, I am considering one of the simplest random dynamical systems that one can consider, and yet I realise that I do not know the answer to one of the most basic questions that one can ask about it!
...

**5**

votes

**0**answers

89 views

### Quantitative approximation of invariant measures by periodic ones

It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures. Is there some knowledge ...

**4**

votes

**0**answers

130 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 $\mathbb{H^2/\Gamma}$ ...

**2**

votes

**0**answers

52 views

### Any minimal WAP dynamical system is distal

I'm trying to show that any minimal WAP dynamical system $(X, G)$ is almost periodic. By Ellis's joint continuity theorem, it suffices to show that any minimal WAP system is distal. There are many ...

**11**

votes

**1**answer

427 views

### Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...

**0**

votes

**1**answer

72 views

### Asymptotically full stationary process

Let $(X_n)_{n \in \mathbb{Z}}$ be a stationary process on a finite set $A$. Say that it is asymptotically full if for every increasing sequence of subsets $B_n \subset A^n$ such that ...

**2**

votes

**0**answers

99 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 ...

**0**

votes

**1**answer

118 views

### Link between Harris recurrence and Ergodicity

Is it possible to obtain Harris recurrent Markov chain from Ergodic chain (in Birkhoff sense) under certain assumption? That is, suppose we know a Markov chain is ergodic (in Birkhoff sense); is it ...

**4**

votes

**2**answers

309 views

### “Typical” convergence rate for the von Neumann mean ergodic theorem

The von-Neumann theorem states that for a unitary operator $U: {\cal H} \mapsto {\cal H}$,
where ${\cal H}$ is a Hilbert space, the following holds:
$$
\lim_{N\to \infty} \frac{1}{N} \sum_{n=1}^N U^n ...

**8**

votes

**2**answers

352 views

### An algorithm for Poincare recurrence time

Define the function $[0,+\infty) \rightarrow R$:
$$ f = \cos (t) + \cos (\sqrt{2} t) + \cos (\sqrt{3} t) + \cos (\sqrt{5} t ) . $$
I want a number $t $ bigger than $10^7$ such that
$$ f(t) > 4 ...

**6**

votes

**1**answer

397 views

### Who introduced the concept of topological mixing?

I am writing an introduction and I want to know who introduced the concept of topological mixing?

**6**

votes

**1**answer

138 views

### Decay of cusps in geometrically finite groups

Let $X=\mathbb{H}^{n}/\Gamma$ be a quotient of hyperbolic space of a geometric finite subgroup. Let $\mu$ be the Bowen-Margulis measure on the unit tangent bundle, and $m$ its projection to $X$.
Fix ...

**5**

votes

**0**answers

264 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. ...

**2**

votes

**2**answers

189 views

### How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...

**2**

votes

**0**answers

75 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 ...

**8**

votes

**3**answers

209 views

### Uniquely ergodicity and polynomial ergodic average

Let $(X,T)$ be a uniquely ergodic system (here X is compact, T is a continuous map form $X$ to itself), so for any continuous function $f:X\rightarrow\mathbb{R}$ we have for any $x\in X$, the ergodic ...

**0**

votes

**0**answers

93 views

### Invariant mesures for expanding maps of the circle

Is there any characterization for the support of T-invariant measures? where T is a C¹ expanding map of the circle i.e. T'(x)>Lambda>1 for all x in the circle.
I know there are periodic and total ...

**9**

votes

**1**answer

134 views

### Nonconventional ergodic averages for commuting transformations

Let $S$ and $T$ be commuting measure-preserving transformations of a standard probability space $(X,\mu)$, so $S$ and $T$ define an action of $\mathbb{Z}^2$ on $(X,\mu)$. I am wondering about ...

**1**

vote

**1**answer

50 views

### Can real-valued Markov processes with continuous surjective sample paths admit a non-trivial “forward-invariant” set?

I have both a more general question (concerning stopping times), and then a more specific application (as described in the title).
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be ...