Questions tagged [ergodic-theory]

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

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WLLN for bootstrap means of stationary ergodic processes?

Setup:$\quad$ Suppose that $(X_n)$ is a stationary ergodic process with $E|X_1|<\infty$. Given $X^{(n)}=(X_1, \dots, X_n)$, select a standard Efron bootstrap subsample $(X_{n,1}^*, \dots, X_{n,m(n)}...
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7 votes
1 answer
153 views

Non-recurrent points of $F(a,b)=(b,ba)$ in a compact metric group $G$

Consider a compact metric group $G$ [A compact topological group $G$ where the topology is generated by an invariant metric]. I am particularly interested in the case where $G$ is the $n$-dimensional ...
5 votes
0 answers
186 views

Direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic

I wonder if there are any direct proof that $g_t=\operatorname{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic (or even stronger,...
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3 votes
0 answers
77 views

Ergodic diffeomorphisms of the circle

From the paper Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889. the following result is known: Let $(E,\Sigma, \mu)$ be a ...
2 votes
0 answers
122 views

Weakly mixing diffeomorphism

From Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889. the following result is known: Let $(E,\Sigma, \mu)$ be a measure ...
1 vote
1 answer
121 views

Ergodicity question

Consider a dynamical system given by the system of ODE. $$\frac{d x_i}{d t} = F_i(\mathbf{x}).$$ It seems to be a well-known fact that this system is ergodic if and only if the kernel of the Koopman ...
4 votes
0 answers
91 views

The Logistic map have subexponential decay of correlation?

I was looking for information about the correlation decay of the logistic map, more precisely if there is any parameter for which its decay is subexponential, in which case I would like to know if it ...
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6 votes
0 answers
72 views

Density of ``diagonal sets'' in amenable groups

Let $G$ be a countable amenable group with a (left) Følner sequence $(F_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F_n)$, in the sense that $$ \lim_{n \to \infty} \...
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1 vote
0 answers
34 views

Oseledets splitting of $f_{0}$ is either dominated with index 1 or trivial at $x$

There is a well-known result by Bochi and Viana that said Let $f_{0} \in \operatorname{Diff}_{\mu}^{1}(M)$ be such that the map $$ f \in \operatorname{Diff}_{\mu}^{1}(M) \mapsto\left(\mathrm{LE}_{1}(...
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6 votes
1 answer
223 views

Are all quasi-regular points on Polish spaces generic points?

Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
1 vote
1 answer
154 views

Using gradient descent in probability case

Suppose we have i.i.d. samples $x_i\sim N(0,\Sigma)$ and $y_i\sim x_i^T\omega^*+\xi_i,\xi_i\sim N(0,1)$ where $\omega^*$ is the fixed point of: $$\omega_{i+1} = \omega_i − \eta\nabla_\omega f(\omega_i,...
2 votes
0 answers
88 views

Uniformly weak mixing transformations

Let $(X, T, \mathcal F, \mu)$ be a nonatomic standard probability space equipped with a measure preserving transformation $T$. We say $T$ is uniformly weak mixing if for every $\varepsilon > 0$, ...
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3 votes
1 answer
112 views

Does uniform recurrence imply uniform convergence of the Birkhoff sums?

Let $(X, T, \mathcal F, \mu)$ be an ergodic measure preserving system with finite measure. Suppose $T$ is uniformly recurrent, in the following sense: For every $A \in \mathcal F$, there exists an $M \...
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4 votes
1 answer
132 views

Ergodic decomposition of the action of a subgroup

Let $G$ be a countable abelian group and let $H \le G$ be a subgroup. Let $G \curvearrowright (X,\mu)$ be an ergodic measure preserving action on some probability space $(X,\mu)$. Now we know that the ...
3 votes
1 answer
87 views

the definition of the topological pressure for matrices

Let $:\Sigma \to GL(d, \mathbb{R})$ be a continuous matrix cocycle over a topologically mixing subshift of finite type $(\Sigma, T)$. We denote by $\Sigma_n$ the set of addmisible words with the ...
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2 votes
2 answers
133 views

Can a diffusion have negative minimum or achieve large value at a given time?

Let $\sigma:\mathbb R_+\times\mathbb R\to [1,2]$ be measurable. Consider the SDE $dX_t = \sigma(t,X_t)dW_t$, where $X_0>0$ is independent of Brownian motion $(W_t)_{t\ge 0}$. For every $T>0$ and ...
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3 votes
0 answers
83 views

What dynamical properties should we expect from systems satisfying statistical ones?

Some results on probability theory can be generalized to more abstract ones in ergodic theory, for example: the strong law of large numbers can be seen as a particular case of Birkhoff's ergodic ...
0 votes
0 answers
48 views

Structure factor for a skew product

Consider the measure preserving system $(\mathbb{T}^2,\mathcal{B}(\mathbb{T}^2),m,T)$ with $m$ being the Haar measure on $\mathbb{T}^2$ and $T$ be defined by $$T(x,y)=(x+\alpha,y+x),\ \text{with}\ \...
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0 votes
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38 views

A non-Kolmogorov system with Lebesgue spectrum: New examples?

It is known that a Kolmogorov system has Lebesgue spectrum, while not every system with Lebesgue spectrum is Kolmogorov. Some of the examples of the latter case are mentioned in Example 9.5.12 of the ...
5 votes
0 answers
140 views

Counterexamples to the Ahlfors measure conjecture in higher dimensions

Let $\Gamma<SO(3,1)$ be a finitely generated, discrete group of isometries of $\mathbb H^3$. By work of Agol, Calegari, Canary, and Gabai, the limit set of $\Gamma$ is either the entire sphere $S^2\...
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5 votes
0 answers
69 views

Lower bound for nonconventional ergodic averages in finite fields

Let $p$ be a sufficiently large prime number and $f\colon\mathbb{F}_{p}\to\mathbb{R}_{\geq 0}$ be a function bounded by 1 such that the average of $f$ over the finite field $\mathbb{F}_{p}$ is at ...
2 votes
1 answer
112 views

Union of admissible words are subshift of finite type

Assume that $Q=(q_{ij})$ is a $k\times k$ with $q_{ij}\in \{0, 1\}.$ The two side subshift of finite type associated to the matrix $Q$ is a left shift map $T:\Sigma_{Q}\rightarrow \Sigma_{Q}$, where ...
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1 vote
1 answer
114 views

Proof that Sturmian shift is uniquely ergodic using irrational rotation

I am finding proofs of unique ergodicity of Sturmian shifts however I want to know if there is a proof that link that to the unique ergodicity of irrational rotations through conjugacy for example or ...
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10 votes
1 answer
325 views

Cohomology for extension problems in symbolic/topological dynamics?

Context: I know essentially nothing about cohomology of any kind, but I have a problem involving classifying obstructions to extensions of certain maps or covers, and I have heard that cohomology is ...
2 votes
1 answer
143 views

In general is $\frac{d\,\mu_1}{d\,\mu_2}\circ T = \frac{d\,T\mu_1}{d\,T\mu_2}$?

Given an ergodic and non-singular dynamic system (definition provided here) $(X, \mathcal{B}, \mu_1, T)$ where $(X, \mathcal{B}, \mu_1)$ is a measure space and $T$ is a fixed transformation, we then ...
2 votes
0 answers
118 views

Choosing the derivative of a flow

I am looking for something like the Franks' Lemma for flows. The celebrated Franks' Lemma states that: Let $f:M \rightarrow M$ be a $C^1$ diffeomorphism and $S=\{p_1,...,p_k\}$ be a finite set of ...
2 votes
0 answers
81 views

A characterization of Shannon entropy in finite sets?

I am trying to solve a complicated probability problem related to Shannon Entropy. Let $(E,p)$ be a finite set with a probability measure $p$ on $E$. $E^n$ is given the probability measure $p^n(x_1, .....
1 vote
0 answers
51 views

Measure preserving system with only trivial eigenfunctions

I want to show that if $(X,\mathcal{X}, \mu, S)$ is a measure preserving system, then $S$ has no non-trivial eigenfunctions if and only if the spectral measures corresponding to all non-constant ...
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3 votes
0 answers
174 views

Complex Hölder space

I already posted this question on math.stackexchange, but got no response and was suggested to post it here. I came across a space in an ergodic theory paper, which I am calling here a (complex) ...
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6 votes
0 answers
93 views

Construction of minimal zero entropy measure-theoretically strong mixing subshift?

Does anyone know of a construction of a subshift (over $\mathbb{Z}$) which is (1) minimal (2) zero (topological) entropy (3) measure-theoretically strong mixing (for some measure)? I am in particular ...
9 votes
0 answers
160 views

For measure-preserving systems, is countable generatability of the invariant $\sigma$-algebra equivalent to almost all points being periodic?

Let $X$ be a second countable Hausdorff topological space, let $T \colon X \to X$ be a Borel-measurable map, define the $\sigma$-algebra $\mathcal{I}=\{A \in \mathcal{B}(X) : T^{-1}(A)=A\}$, and for ...
1 vote
0 answers
109 views

Relation between the distance projective maps and their angles

Let $f:N \to \mathbb{R}^2$ be a differentiable map of smooth manifolds. Let $\mathbb{R}^2$ be decomposed as a direct sum of line bundles, i.e. $\mathbb{R}^2=E(x) \oplus F(x)$, where $F(x)$ and $E(x)$ ...
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8 votes
1 answer
193 views

Equivalent definitions of topological weak mixing

A dynamical system $f:X\to X$ is said to be topologically transitive if for any two nonempty open sets $U,V$ there exists $n \in \mathbb{Z}$ such that $f^{\circ n}(U) \cap V \neq \emptyset$. The ...
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3 votes
1 answer
119 views

Extension of Khintchine's recurrence in a simple case

Suppose an ergodic system $(X,\mathcal{B},\mu,T)$ has a Kronecker factor that is isomorphic to an ergodic rotation, say on the Torus. How can one prove that the large intersection property holds for $...
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2 votes
1 answer
305 views

Does mixing automatically imply this seemingly stronger "uniform modulo re-ordering" version of mixing?

THE QUESTION Let $(X,\mathcal{X})$ be a standard Borel space, $T \colon X \to X$ a measurable map, and $\mu$ a $T$-mixing probability measure. Is it necessarily the case that for all $A \in \mathcal{...
4 votes
0 answers
53 views

Ergodic transformations, their Poisson suspensions and their Krieger types

Let $T$ be an ergodic nonsingular tranformation of a Lebesgue space. Suppose that the Poisson transformation $T^*$ of $T$ is well-defined and ergodic. Denote by $\alpha$ the Krieger type of $T$ and by ...
4 votes
1 answer
183 views

Maximal ergodic inequality

A map $f: X \to X$ preserves an ergodic probability $\mu$, i.e., $\mu \circ f^{-1}=\mu$ and for any $\phi: X \to \mathbb{R}$ with $\int \phi d\mu=0$, $$\frac{1}{n} \sum_{i \le n} \phi \circ f^i \to 0 \...
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2 votes
2 answers
175 views

Measure preserving transformation that makes two partitions independent

I am looking for a reference for the following result. I think it is well known but I haven't seen it written down anywhere. Let $(X, \mathcal{B}, \mu)$ be a standard measure space and let $\mathcal{...
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6 votes
1 answer
127 views

Mañé's example of an attractor with no natural measure

I'm reading Milnor's notes on dynamical systems and in Lecture 3 he gives an example of an attractor with no natural measure, which he attributes to Mañé. I can find no other reference in which this ...
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2 votes
0 answers
94 views

Almost periodic functions in weak mixing extension

In Theorem 3.4.6 of the 'Lecture notes on ergodic theory' by Jesse Peterson, it is shown that in a weak mixing extension, every almost periodic function is trivial. I have a doubt in the proof of this ...
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0 votes
1 answer
115 views

Entropy maximising ergodic transformation

Let $(\Omega, \mathcal F, \mu)$ be a standard probability space. Question: For each $f \in L^\infty (\Omega)$, does there exist an ergodic measure preserving transformation $T: \Omega \to \Omega$ such ...
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6 votes
1 answer
155 views

SRB measure and Gibbs u-state

I have been reading the famous paper of Alves, Bonatti, and Viana where they proved that there is an SRB measure for partially hyperbolic systems. Since I am new to this field, I have some basic ...
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2 votes
1 answer
247 views

Diophantine equations and ergodic theorems

In the paper by Akos Magyar, Diophantine Equations and Ergodic Theorems, one states in page 923 the following theorem: Theorem 1: Let $Q(m)$ be a nondegenerate polynomial and $\Lambda$ is ...
3 votes
0 answers
183 views

The baker problem

Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any subset $A$ of $S$ let $A^{c}$ denote its complement in $S$, and $\overline{A}$ its closure in $S$. Given a measurable map $g: W \...
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6 votes
2 answers
338 views

3-periodic point implies positive topological entropy

When I learn some basic ergodic theory, I encounter an interesting exercise. As we all know, 3-periodic point often means chaos. Therefore, when a continuous map has a 3-periodic point, it may have ...
2 votes
1 answer
107 views

K-flows reference

The following paper is about how a K-flow is produced from a K-induced map, but it is written in Russian. Does someone know where to find its English version? Do some textbooks include this topic? B. ...
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3 votes
2 answers
103 views

The closure of the orbit of an irrational grid contains the fiber

Let $G=\text{SL}(d,\mathbb R)$ and $\Gamma = \text{SL}(d,\mathbb Z)$. The homogeneous space is identified with the space of unimodular lattices, denoted $X_d$. Let $Y_d$ denote the space of unimodular ...
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5 votes
1 answer
121 views

Recurrence of ergodic processes

Let $(X_1,X_2,\ldots)$ be a stationary ergodic process with each $X_n$ a real random variable taking values in $[-1,+1]$. Suppose that $\mathbb{E}[X_n]=0$. Let $S_n = \sum_{k=1}^n X_k$. Is the process ...
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2 votes
1 answer
268 views

Is the consecutive sum set large in general?

$\DeclareMathOperator\CSS{CSS}$It is well known that for a set $A$ of integers, if $\gcd(A) = d$, then the set of (integer) linear combinations of $A$ is $d\mathbb{Z}$. I'm looking for a probability ...
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0 votes
0 answers
79 views

Length of walking on a graph

Given a finite directed connected graph $G$, let $P_{circle}$ be the set of finitely long circle paths on $G$ (a circle path is a path with identical starting and ending vertex). It is well known that ...
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