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

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A rigorous proof of this conjecture?

I was thinking in this question, after a failed success studying certain sequences of sets (I thank Emil Jeřábek and Ian Morris for pointing me this out). Inspired by this, I wrote a formal proof of ...
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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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66 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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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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22 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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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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1answer
114 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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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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124 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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193 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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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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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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102 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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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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319 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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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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202 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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117 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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166 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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2answers
101 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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115 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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287 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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130 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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598 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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134 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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292 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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130 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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1answer
83 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 ...
4
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1answer
154 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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96 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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192 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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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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38 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 ...
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Circular process ergodic?

Let us define a continuous-time Markov process on a circle consisting of $m-$ equally spaced points, i.e. every point has two neighbours. Now, we define a space of functions $S:= ...
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89 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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144 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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88 views

Shift Invariance of Backward Martingales for tail trivial probability measures

Consider the infinite cartesian product $\Omega=\{0,1\}^{\mathbb{N}}$ as a measurable space endowed with the $\sigma$-algebra $\mathscr{F}$ generated by the cylinder sets and $\sigma:\Omega\to\Omega$ ...
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235 views

Estimate of the number of rabbit integers with a given congruence

Consider the Fibonacci words $B_n$: $B_1 = 1$ $B_2 = 10$ $B_3 = 101$ $B_4 = 10110$ $B_5 = 10110101$ (start with $B_1=1$, and go from $B_n$ to $B_{n+1}$ by replacing every occurence of $1$ in ...
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130 views

Is there a mixing condition to get the decay property I want?

Let $(X,\mu)$ be a probability measure space and $T:X\to X$ an ergodic invertible measure preserving transformation. Consider a measurable set $A\subset X$ with $0<\mu(A)<1$ For each $N$ define ...
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Strict factor of a dynamical system with the same entropy [closed]

Say that a factor of an invertible measure-preserving transformation $T$ is strict if it is not isomorphic to $T$. Does there exist an invertibe mpt $T$ such that $0 < h(T) < \infty$ and ...
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Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action

Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...
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124 views

The scope of correspondence principle in quantum chaos

My understanding of the so-called correspondence principle in quantum chaos, is that it is a connection between the behaviour of a classical Hamiltonian system (chaotic/completely integrable) and the ...
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121 views

Generator determined by finitely many translates implies zero entropy

Let $T$ be a measure preserving transformation of a standard probability space $(X,\mathcal{B},\mu)$. A partition $\alpha$ of $X$ is said to be a generator for $T$ if the smallest $T$ invariant ...
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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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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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1answer
79 views

Two-side deviations for ergodic sums

Let $(X,\mu)$ be a probability space and $f\colon (X,\mu)\to (X,\mu)$ be an ergodic automorphism. Let $\phi\in L^\infty(X,\mu)$ be such that $\int\phi d\mu=0$. Suppose that for $\mu$-a.e. $x\in X$, ...
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Almost sure convergence of double nonconventional ergodic averages with respect to $L^p$ function

A famous result of J. Bourgain says that for a probability measure preserving system $(X,\beta,\mu,T)$, with $T_1$ and $T_2$ powers of $T$, we have that for $f_1$, $f_2\in ...
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Characterizing when matrices are 'dissipative'

An $n$ by $n$ matrix A is said to be dissipative with respect to a norm $\|\cdot \|$ if for all $x$ and $t\geq 0$, we have $\|e^{At}x\|\leq\|x\|$. Two matrices $A$ and $B$ are said to be jointly ...
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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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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 ...