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

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95 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 ...

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**1**answer

455 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**

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**1**answer

165 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 ...

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89 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 ...

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**1**answer

82 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)) = ...

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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?
...

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111 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 ...

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112 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 ...

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380 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, ...

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votes

**1**answer

91 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 ...

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38 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 ...

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666 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 ...

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120 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 ...

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**1**answer

101 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 ...

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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 ...

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**1**answer

632 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 ...

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103 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 ...

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70 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 ...

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125 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 ...

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votes

**1**answer

119 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} ...

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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 ...

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83 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 ...

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**1**answer

254 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: ...

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**1**answer

570 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 ...

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78 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 ...

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103 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 ...

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94 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 ...

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590 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 ...

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**1**answer

204 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 ...

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**0**answers

68 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
...

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votes

**1**answer

205 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 $(X,\Gamma)$ ...

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**1**answer

119 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?
...

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

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169 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 ...

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**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 ...

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290 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? ...

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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 ...

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132 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 ...

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**1**answer

523 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 ...

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votes

**1**answer

136 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.

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**1**answer

299 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) = ...

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131 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 $N|I|$ ...

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**1**answer

85 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 ...

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**1**answer

157 views

### Discrete spectrum and almost periodicity

According to Vershik, an ergodic invertible measure-preserving transformation $T$ on a Lebesgue space $X$ has discrete spectrum if and only if for every bounded measurable function $f\colon X \to ...

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**3**answers

243 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 ...

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146 views

### Transformation extending all ergodic rotations

Is there an invertible measure-preserving transformation (preferably a nice one) admitting every irrational rotation as a factor ? I guess the spectrum is the relevant tool to address this question ...

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**1**answer

196 views

### What are the generating partitions of the odometer?

According to the countable generator theorem, every ergodic invertible measure-preserving transformation has a generating partition.
What are the generating partitions of the dyadic odometer ? I ...

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93 views

### weak-* versus entropy growth

General question. Let $\eta_{n}$ be a sequence of invariant measures on $\{0,1,2,...,p-1\}^{\mathbb{N}}$ and $B$ the Bernoulli uniform measure. Knowing that $\eta_{n} \rightarrow B$ in the weak-* ...

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92 views

### Transformations whose product with a given rotation are ergodic

I am interested in the ergodic (invertible) transformations $T$ such that $T\times R_\theta$ is ergodic where $R_\theta$ is the rotation on $S^1$ with a given irrational angle $\theta$ (not all ...

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40 views

### Transformations whose product with the odometer are ergodic

Let $T$ be an invertible ergodic transformation on a Lebesgue space $X$ and $O$ be the dyadic odometer on $(0,1)$. Is it true that $T\times O$ is ergodic if and only if $T^{2^n}$ is ergodic for every ...