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

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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 ...
3
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3answers
233 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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68 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 ...
3
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1answer
138 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 ...
4
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1answer
175 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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73 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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74 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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515 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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33 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 ...
3
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75 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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52 views

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:= ...
8
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1answer
314 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 ...
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205 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 $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...
2
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1answer
85 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 $\sigma:\Omega\to\Omega$ ...
2
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1answer
230 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 ...
8
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117 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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360 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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233 views

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 ...
4
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1answer
126 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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90 views

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 ...
6
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1answer
114 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 ...
5
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592 views

Poincaré Recurrence and Dense Sets

This is kind of a spin-off of the question asked here. Take the interval $X:=[0,1]$ with $\mu$ being standard Lebesgue measure. Let $f$ be a measure preserving map $f:[0,1]\rightarrow [0,1]$. The ...
4
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1answer
204 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 $x$ ...
6
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2answers
331 views

Ruelle inequality on a noncompact space

Does someone have a reference where the Ruelle inequality would be proved in the following context. Let $M$ be a non compact smooth manifold, and $f:M\to M$ be a $C^1$-diffeomorphism (or $C^2$, ...
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432 views

What is known about dynamics on Grassmannians?

I have found myself becoming interested in dynamical systems given by homeomorphisms acting on $G(r,d)$, the space of $r$-dimensional subspaces of $\mathbb{R}^d$. I tried to do a literature search ...
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209 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 ...
4
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1answer
77 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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60 views

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 ...
7
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2answers
148 views

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 ...
7
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1answer
85 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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187 views

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 ...
5
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3answers
280 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 ...
5
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1answer
224 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 ...
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155 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 ...
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2answers
625 views

Is this ergodic inequality true?

Is anything similar to the following inequality true, $\displaystyle P\{\max_{n \leq k \leq m} |A_k f - A_n f| > \epsilon\} \leq C \frac{||A_m f - A_n f||_1}{\epsilon}$ where $A_n f = ...
2
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2answers
95 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 ...
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179 views

Express measurable entropy in terms of Fourier coefficients of the measure

Let $S^1$ be the unit circle and $T:S^1\to S^1$ be a continuous map. Suppose $\mu$ is a $T$-invariant Borel probability measure on $S^1$, that is, $\mu(T^{-1}A)=\mu(A)$ for every Borel subset $A$ of ...
0
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1answer
124 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 ...
0
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1answer
109 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 ...
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1answer
104 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 $\bar{d}$ ...
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2answers
598 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 $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...
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82 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
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1answer
118 views

Halmos recurrence theorem for a locally compact group

The recurrence theorem of Halmos is well known in the case of a non-singular endomorphism $T$ of a measured space $(X,\mathcal B,\mu)$. A measurable subset $A$ is contained in the conservative part ...
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244 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} = ...
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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 ...
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1answer
104 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 ...
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56 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 ...
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135 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 ...
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202 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 $\Gamma$ ...
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100 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) ...