# Questions tagged [ergodic-theory]

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

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questions

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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)}...

7
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### 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 ...

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### 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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### 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 ...

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### 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
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1
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### 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 ...

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### 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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### 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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### 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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### 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 $...

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1
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154
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### 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,...

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### 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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### 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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### 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 ...

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### 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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### 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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### 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 ...

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### 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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### 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 ...

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### 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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### 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 ...

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### 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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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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, .....

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### 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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### 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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### 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 ...

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### 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 ...

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### 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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### 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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### 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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### 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{...

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### 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 ...

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### 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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### 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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### 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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### 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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### 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 ...

6
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155
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### 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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### 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 ...

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### 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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### 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 ...

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### 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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103
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### 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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### 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 ...

2
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268
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### 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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### 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 ...