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

**0**

votes

**1**answer

32 views

### A question on invariant measures

Let $(X, \mathcal{B}, T)$ be a topological dynamical system and $M(X, T)$ be the set of all invariant measures.
I do not know is there some nice functional characterization of the following set
$\{...

**19**

votes

**1**answer

703 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 ...

**-1**

votes

**0**answers

77 views

### Are the theorems of Ergodic Theory valid for non-probability spaces?

The theorems in Ergodic Theory have assumed a probability measure, always. I am interested to know if they hold even when the space is not equipped with a probability measure. In other words, if my ...

**5**

votes

**2**answers

83 views

### Normalization in Birkhoff's theorem and its failure in infinite measure spaces

I have two questions, somehow related. The first one, has to do with the Birkhoff's ergodic theorem. In its classical formulation, it states that if we have a probability space $(X,\mathcal{B},\mu)$ ...

**12**

votes

**1**answer

486 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 ...

**3**

votes

**1**answer

97 views

### On the spectrum of stationary Gaussian process

What is the condition for ergodicity, weakly mixing, and strongly mixing properties of Gaussian process in terms of its spectrum?
In a similar way let us consider a stationary vector valued Gaussian ...

**1**

vote

**1**answer

56 views

### Uniqueness of invariant measure for equivalent transition probabilities

Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \...

**6**

votes

**1**answer

187 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 almost ...

**0**

votes

**0**answers

74 views

### Orbit intersection in toral automorphisms

Let $f:(\mathbb{R/Z})^2\to (\mathbb{R/Z})^2$ be a hyperbolic toral automorphism induced by a matrix $A\in GL_2(\mathbb{Z})$. Consider a measurable (wrt to the Lebesgue measure $\mu$ on the torus) ...

**1**

vote

**0**answers

54 views

### Converge of measures [closed]

Good night, we have the following:
Let $(X,d)$ is a general metric space, $\mathcal{M}(Y)$ is the set of finite Borel measures on $Y$ and $C_B(Y)$ denotes the Banach space of bounded continuous ...

**1**

vote

**0**answers

108 views

### What interesting information can be deduced from knowledge of how deep a geodesic ventures into the cusp

First of all I have to apologise as I am not a geometer and my knowledge of geometry is poor. Let $M$ be the modular surface and $\gamma$ to denote a geodesic in $M$. In the the following paper by ...

**2**

votes

**0**answers

126 views

### markov processes and ergodic theory

For an ergodic Markov Chain
$$
\frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f]
$$
where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...

**6**

votes

**1**answer

94 views

### Is there a complete Riemannian manifold with infinite volume whose the time-one map of the geosesic flow is recurrent?

Let M be complete Riemannian manifold M with infinite volume, it is know that the geodesic flow, $\varphi^t:T^1M \rightarrow T^1M$ preserves the Liouville measure $\mu$, that is, $\mu(\varphi^t(A)) = ...

**0**

votes

**0**answers

22 views

### Groups with many non-conjugate but orbit equivalent actions

Which countable discrete groups (apart from the amenable ones) admit uncountably many mutually non-conjugate free ergodic probability measure preserving actions that are all mutually orbit equivalent?

**6**

votes

**2**answers

121 views

### Relationship between Multiplicative Ergodic Theorems

One version of Oseledets' Multiplicative Ergodic Theorem states that if $\sigma$ is an ergodic measure-preserving transformation of a space $(\Omega,\mathbb P)$ and if $A\colon\Omega\to GL(d,\mathbb R)...

**4**

votes

**0**answers

116 views

### Why do we care about simplicity of the spectrum in Oseledets' theorem?

Oseledets' theorem is a fundamental result in Ergodic theory (see for example here, or Chapter 4 of Lectures on Lyapunov Exponents by Marcelo Viana).
The simplicity of the spectrum has been studied ...

**12**

votes

**2**answers

408 views

### Nuclear operators/spaces and transfer operators

While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...

**4**

votes

**1**answer

101 views

### Ergodic, non-atomic measure on the circle which are $\times 2$ and $\times \frac12$ invariant

There any many ergodic, $T$-invariant, non-atomic measures on the space $X = [0,1)$, where $Tx = 2x \pmod 1$ is the doubling map.
My question is: are any such measures also $T^{-1}$-invariant? BYO ...

**0**

votes

**0**answers

41 views

### Necessity of expansiveness for existence absolutely continuous invariant measures for piecewise smooth maps of an interval

A map $\tau:[0,1]\to[0,1]$ is piecewise smooth (or $C^r$) if there is a partition of $[0,1]$ into intervals, $[0,1]=\cup I_n$, (which can be either finite or countable) such that the restriction of $\...

**2**

votes

**0**answers

708 views

### What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required.
In random signal theory, this distribution is typically a ...

**7**

votes

**2**answers

127 views

### List of Bernoulli chaotic systems

Which discrete chaotic systems are known to be Bernoulli (i.e. measure theoretically isomorphic to a Bernoulli shift, one-sided or two-sided)?
I am aware that it is known for some uniformly ...

**5**

votes

**1**answer

105 views

### General properties of the Ruelle operator

Recently I have read Parry and Pollicott's book, Zeta functions and the periodic orbit structure of hyperbolic dynamics.
I have been interested in some technical properties of the Ruelle-Perron-...

**0**

votes

**0**answers

24 views

### Equivalence classes on an ordered Bratteli diagram

Let $S$ be the adic transformation preserving a probability measure $\mu$ on the set $\Gamma$ of infinite paths of a $\mathbb{N}$-graded ordered Bratteli graph.
For every $n \geq 0$ define the ...

**2**

votes

**2**answers

104 views

### Invariant $\sigma$-field of a product with a weakly mixing transformation

It is known that an invertible mpt $S$ is weakly mixing if and only if $S \times T$ is ergodic for any ergodic invertible mpt $T$. Is it more generally true that the invariant $\sigma$-field of $S \...

**1**

vote

**0**answers

76 views

### Is there an area preserving toral endomorphism with critical point?

An endomorphism is a continuous map $f:\mathbb{T}^2 \to \mathbb{T}^2$. An conservative endomorphism is an endomorphism that is area preserving $(m(f^{-1}(U)=m(U), \forall U$ borel set and m is the ...

**3**

votes

**2**answers

128 views

### Hausdorff dimension of sequence space

Let $\Omega =\{0,1\}^{\mathbb{N}}$ denote the set of infinite sequences with elements $0$ or $1$. Let $d$ be the metric on $\Omega$ given by $d((x_n),(y_n))=1/2^m$, where $m=\min\{i\in\mathbb{N}\,:\,...

**0**

votes

**1**answer

124 views

### Weighted sum of i.i.d. random variables

Suppose you have a positive sequence $X_1,X_2,\dots$ of i.i.d. random variables with the property that
$$
\mathbb{E}[\log(X_1)]<\infty.
$$
Is it true that
$$
\limsup_{n\to\infty} e^{-n}\sum_{k=1}^...

**2**

votes

**2**answers

196 views

### How to generalize normal number theorem

The Borel number theorem states that with respect to Lebesgue measure, almost all real numbers are normal numbers. It is sometimes stated in the context of the compact interval $[0,1]$, where one ...

**1**

vote

**0**answers

85 views

### Ergodic skew product on $\mathbb T^d\times U(2)$

Let $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ be the $d$-dimensional torus with normalized Haar measure $\mu_1$ and let $U(2)$ be the group of $2\times2$ unitary matrices with normalized Haar measure $\...

**10**

votes

**1**answer

255 views

### About positive upper density

For $S\subset \mathbb{N}$ define the upper density as $D^{\ast
}(S)=\limsup_{n\rightarrow \infty }\frac{\left\vert S\cap \{1,2,\ldots,n\} \right\vert }{%
\left\vert n\right\vert }.$
Question: ...

**8**

votes

**1**answer

572 views

### Sort-of Converse of Kolmogorov Zero-One Theorem

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov Zero-One Theorem states that
Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in \...

**0**

votes

**0**answers

79 views

### applications of ergodic theory to periodicity of regular continued-fractions

The usual application one sees of ergodic theory to the regular continued-fractions is the Gauss-Kuzmin Theorem on the frequency of positive integers in the continued fraction expansion for almost all ...

**4**

votes

**2**answers

105 views

### Sturmian subword whose reverse is not a subword

Let ${\cal L}_n$ be the set of all subwords of length $n$ of a biinfinite Sturmian sequence, induced by a rotation coding with irrational angle $\theta$.
Take a word $w \in {\cal L}_{2^n}$ and write ...

**6**

votes

**2**answers

96 views

### Question about a certain coding of rotations

Notation: A word $w$ on the alphabet $A=\{a,b\}$ having $2p$ letters can be viewed as a word $w'$ having $p$ letters on the alphabet $A'=A^2$. I denote by $\beta(w)$ the number of occurences of the ...

**6**

votes

**2**answers

608 views

### Can ergodic theory help to prove ergodicity of general Markov chain?

I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking ...

**3**

votes

**1**answer

206 views

### Pointwise ergodic theorem for amenable semigroups

Using tempered Følner 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 ...

**2**

votes

**0**answers

70 views

### Question about martin boundaries of random walks induced on transient subgroups

Suppose $\Gamma$ is a discrete, finitely generated, non-amenable group, and
consider a random walk given by a measure $\mu$.
Assume the measure is symmetric, finitely generated, and the support of
$\...

**3**

votes

**1**answer

206 views

### Examples of topological dynamical systems with countably infinitely many ergodic invariant measures

Suppose a discrete group $\Gamma$ acts on a connected compact metrizable space $X$ by homeomorphisms. Denote such a topological dynamical system by $(X,\Gamma)$.
Question: is there any $(X,\Gamma)$ ...

**1**

vote

**1**answer

123 views

### Neat definition of Harris Ergodicity

I can't find any reference where the definition of Harris Ergodicity for Continuous time Markov processes is defined.
a) What would be exactly the definition?
b) What reference could be helpful?
...

**2**

votes

**2**answers

304 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 $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...

**3**

votes

**0**answers

172 views

### Repartition of 1's in the “Chacon word”

Consider the "Chacon words": $B_0=0$ and $B_{n+1} = B_nB_n1B_n$. The word $B_n$ has $\ell_n := \frac{3^{n+1}-1}{2}$ digits and the number of $1$'s in $B_n$ is $\ell_n - 3^n = \ell_{n-1} \sim \ell_n/3$...

**6**

votes

**1**answer

117 views

### Can a smooth diffeomorphism of a Riemannian manifold have only positive Lyapunov exponents on a large set?

Let $M$ be a compact Riemannian manifold, $f: M \to M$ a diffeomorphism, and $\mu$ an ergodic measure for $M$. Suppose that the support of $\mu$ is not a finite set. Is it possible that all the ...

**5**

votes

**2**answers

294 views

### Can a smooth diffeomorphisms of a Riemannian manifold have only positive Lyapunov exponents?

Let $\mu$ be some ergodic measure of our compact Riemannian manifold $M$, which is preserved by $f\in Diff^{1+\beta}(M)$. Is it possible that all the Lyapunov exponents of $\mu$ will be positive? ...

**4**

votes

**2**answers

127 views

### Iterative renormalizations of a rotation

(Underlying job: I am trying to construct an adic representation of a rotation.)
The question involves an iterative construction. At step $n$, one constructs a partition $P_n$ of $(0,1)$ and a map $\...

**3**

votes

**2**answers

135 views

### Convex combinations of Bernoulli Measures

How big is the weak-* closure of the set of all (finite) convex combinations of Bernoulli measures among all invariant probability measures?
I mean, we are in the symbolic space $\{1,2,\ldots,d\}^{\...

**7**

votes

**1**answer

534 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 ...

**4**

votes

**1**answer

142 views

### Uniquely ergodic and strongly mixing transformation

Is there an example of a non-trivial measure preserving transformation that is uniquely ergodic and strongly mixing (in the measure theoretic sense)? This was asked here, but with no answer.

**19**

votes

**1**answer

309 views

### Possible limits of $(1/n) \sum_{k=0}^{n-1} e^{i2\pi \cdot 2^k\alpha}$

I made a throwaway comment on math stackexchange the other day that got me thinking about the following question. Let
$$ f_n (\alpha) = \frac1n \sum_{k=0}^{n-1} e(2^k\alpha),$$
where $e(x) = \exp(i2\...

**1**

vote

**0**answers

136 views

### A weighted ergodic average

According to my simulations, it looks like the number of times that the $N$ first iterates $u_0$, $\ldots$, $u_{N-1}$ of the sequence $(u_n)$ defined here meets an interval $I$ is close to $N|I|$ ...

**1**

vote

**1**answer

86 views

### Ergodicity of elementary symmetric polynomials with noncommutable variables

Let $\{X_n\}$ be an ergodic sequence of random variables, $X_n:(\Omega,\mathcal{F})\to (S,\mathcal{S})$ where the target set $S$ is a matrix ring. My question is,
Can the following limit be found ...