Questions tagged [ergodic-theory]

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

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Fixed point subalgebra

Suppose that $M$ is a von Neuman algebra and we have an action of a finite group $G$ on $M$. Denote by $M^{G}$ the fixed point subalgebra and suppose that $M^{G}=\mathbb{C}$ (i.e., we have an ergodic ...
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Almost every $m\times n$ real matrix is Dirichlet approximable

Let $\| \cdot \|$ denote the maximum norm in Euclidean spaces. Consider the set $D_{m,n}$ of $m \times n $ real matrices satisfying that the system of inequalities $$\|Aq-p\|^m < \frac{1}{T}, \|q\|^...
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1answer
152 views

Existence of a continuous ergodic dynamical system for a given distribution?

It seems to me that given a distribution (which is well-behaved), there should be at least an ergodic dynamical system that its time average would create this distribution. Is this question already ...
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3answers
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Examples in ergodic theory and topological dynamics

I am currently studying basic ergodic theory: Invariant measures Poincaré recurrence theorem Invariant measure for continuous transformations The ergodic theorems and applications Ergodic ...
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1answer
125 views

The mean ergodic theorem for weakly mixing extension

I asked this question in https://math.stackexchange.com/q/4236870/528430, but did not get any help. I got stuck with the following while going through the proof of Lemma 3.21 from the book 'Ergodic ...
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Ergodic action on product spaces

Let $(X_1 \times X_2,d\mu)$ be a measure space with $X_2$ compact. Suppose that we have a continuous (diagonal) action of a topological group $G$ on $X=X_1 \times X_2$. I know that the action of $G$ ...
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1answer
203 views

A property of rapid sequences of natural numbers

$\newcommand{\IR}{\mathbb R}$ $\newcommand{\IT}{\mathbb T}$ $\newcommand{\w}{\omega}$ $\newcommand{\e}{\varepsilon}$ Taras Banakh and me proceed a long quest answering a question of ougao at ...
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1answer
245 views

Asymptotically invariant maps and strongly ergodic actions

Let $\Gamma$ be a countable group which acts strongly ergodically on a probability measure space $(X,\mu)$. Let $\sigma_k:X \rightarrow Y$ be a sequence of measurable functions into a complete metric ...
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30 views

Under reasonable assumptions, is a closed invariant graph with only negative Lyapunov exponents necessarily stable?

Let $\Omega$ and $M$ be compact $C^\infty$ manifolds, let $\theta \colon \Omega \to \Omega$ be a $C^\infty$ diffeomorphism, and let $\Theta \colon \Omega \times M \to \Omega \times M$ be a $C^\infty$ ...
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122 views

Uniform distribution modulo 1 and probability [closed]

Define counting function $A(E; N; \omega)$ as the number of terms $x_n, 1\leq n\leq N$, for which $\{x_n\}\in E$. Then the sequence $\omega=(x_n), n=1,2,...,$ of real numbers is said to be uniformly ...
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1answer
365 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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116 views

Friedland metric entropy

I was asking if it is possible to extend the definition of topological Friedland entropy for $\mathbb{Z}^d$ continuos actions to measure preserving actions. The topologica Friedland entropy is ...
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142 views

Ergodicity of induced system

Suppose $(X,\mathcal{F},\mu,T)$ is an ergodic measure preserving dynamical system. Let $Y\subset X$ be such that $\mu(Y)>0$ and suppose there is an integrable function $R:Y\to \mathbb{N}$ such that ...
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1answer
123 views

Does the following condition imply ergodicity?

Let $(X,\mathcal F,μ,T)$ be a dynamical system (i.e. μ is a probability measure and Τ is μ-preserving) and $\mathcal S\subset\mathcal F$ be a family of sets such that for any $A \in \mathcal F$ and $ε&...
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1answer
78 views

Exponential mixing for subshifts

I asked this question on Math.StackExchange some time ago and got no responses. Let $G=(V,E)$ be a finite graph with adjacency matrix $A$. Let us consider the associated subshift of finite type $$ \...
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2answers
799 views

Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

I've posted this question on Math Stack Exchange, but I want to bring it here too, because 1) the proof seems missing in the literature, although they are some sporadic mentions and 2) maybe it ...
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1answer
335 views

How to analytically prove chaos

Consider the following map \begin{align*} T \colon \mathbb{R}\times\mathbb{S}^1 \to & \mathbb{R}\times\mathbb{S}^1 \\ (x,\theta) \mapsto & \left(\frac{x}{4}+ \sin^2\left(\pi\left(\theta+\...
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60 views

Disjointness of processes obtained from "cutting and stacking"

Two ergodic probability measure-preserving systems in ergodic theory, $T$ of $(X,\mu)$ and $S$ of $(Y,\nu)$, are said to be disjoint if the only joining (i.e. $T\times S$-invariant measure on $X\times ...
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3answers
370 views

Count of non-trivial ergodic measures of a topological dynamical system

Given a compact Hausdorff space $X$ and a continuous mapping $\varphi: X \to X$. We denote by $C(X)$ the space of continuous functions $f: X \to \mathbb{C}$. A probability measure $\mu$ on the Borel-$\...
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1answer
188 views

Does an “almost weakly mixing” transformation admit a non-null ergodic component?

Problem set up: Let $\mathbf X := (X, \mathcal A, \mu)$ be a standard probability space. We say that a measure preserving transformation $T$ on $\mathbf X$ is $\varepsilon$-almost weakly mixing if for ...
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1answer
173 views

Irrational rotations are rank 2 by intervals without spacers

Let $\alpha$ be an irrational number, and $R_\alpha$ be the rotation by $\alpha$, that is $R_\alpha(x)=x+\alpha\bmod 1$. S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. ...
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Correspondence between Hoelder cocycles and Hoelder potential functions for noncompact negatively curved manifolds

Let $\tilde{M}$ be the universal cover of a pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary. When $M$...
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3answers
371 views

All two-point correlations equal to $0$, three-point correlation not $0$?

Let $a_1,a_2,a_3,\dotsc \in \{-1,1\}$ be a sequence. Suppose that, for all $j>0$ and all $\epsilon, \epsilon'\in \{-1,1\}$, the proportion of $n\geq 1$ such that $(a_n,a_{n+j}) = (\epsilon,\epsilon'...
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3answers
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Connection between properties of dynamical and ergodic systems

While studying topological and ergodic dynamics, I've got quite perplexed by the different properties a system might have (minimality, regionally recurring, transitivity, mixing, ergodic, uniquely ...
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1answer
398 views

Picking a representative in a continuous way

I'm hoping for some ideas/pointers here. I'm experimenting with a Livschitz theorem for functions on a locally compact Abelian group, where the periodic orbit sums take values in a closed subgroup. ...
3
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1answer
121 views

Does full shift have the local product structure?

We say that an invariant measure $\mu$ on some symbolic space $\Sigma$ has local product structure if there is a measurable function $\psi: \Sigma \rightarrow(0, \infty)$ such that the restriction is ...
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1answer
185 views

A sufficient condition for an ergodic system to be weakly mixing

Let $\mathbf X := (X, \mathcal S, \mu, T)$ be an ergodic measure preserving system with finite measure such that for every increasing sequence $\{n_k\}$ of natural numbers with positive lower density, ...
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0answers
145 views

Approximating rotations on a torus with irrational rotations

Consider a rotation of the form $x\mapsto e^{i\theta}x$, for $x$ on the unit circle. By iterating this rotation, one can approximate any other rotation $x\mapsto e^{i\phi}x$ arbitrarily well, as long ...
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1answer
107 views

Random sequence with positive Lyapunov exponent?

Consider the following self-adjoint matrix $A_X = \begin{pmatrix} 0 & -i \\ i & X \end{pmatrix},$ where $i$ is the imaginary unit and $X$ is a uniformly distributed random variable on some ...
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1answer
172 views

Robustness of ergodic dynamical systems

Let $\mathbf X := (X, \mathcal F, \mu)$ be a standard probability space. For an ergodic measure preserving transformation $T$, we define the ergodic robustness $\mathcal R(T)$ of $T$ as follows: For $...
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140 views

Ergodic Theory and Euler-Mascheroni Constant

I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or $\zeta(5)$. A professor guided me that arithmetic nature of constants are a ...
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1answer
119 views

Is a “uniformly minimal” dynamical system ergodic?

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, measure preserving and uniformly transitive in the sense ...
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70 views

Is a “uniformly syndetic” dynamical system weak mixing?

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, measure preserving and uniformly syndetic in the sense that ...
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1answer
173 views

Smooth dynamics with zero Lyapunov exponents

Apologies if this is a vague question. It seems that a lot of the literature in smooth dynamics is focused on understanding systems that exhibit hyperbolic/non-uniformly hyperbolic behavior. In other ...
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1answer
74 views

(Exponential) Mixing property for Gauss map - going from cylinders to intervals

I'm trying to understand the proof of a mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step. The Gauss map $T$, ...
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89 views

Quotient measure on locally compact spaces

Suppose we are given a locally compact topological space $X$ and a discreet group $G$ acting on it (we can assume the action to be proper). Given a Radon probability measure on the quotient space $G \...
2
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1answer
93 views

positive of the largest Lyapunov exponent

Let $\alpha\in \mathbb{R} / \mathbb{Q}$, \begin{equation} A(x)=\left(\begin{array}{ll} \frac{1}{{\lambda}^2}-2 \cos 2\pi x -1& 2\lambda \cos 2\pi x-\frac{1}{{\lambda}} \\ \frac{1}{{\lambda}} &...
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1answer
113 views

Ergodic theorem on limit of periodic transformations?

Suppose $(X,\mu)$ is a probability space, and $T_n, n \in \mathbb N$, is a sequence of periodic measure preserving transformations. For $x \in X$ and $f : X \to \mathbb R$, let $\mathrm{avg}_{f,n}(x)$...
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1answer
200 views

Does an “almost mixing” transformation admit a non-null ergodic component?

Problem set up: Let $\mathbf X := (X, \mathcal A, \mu)$ be a standard probability space. We say that a measure preserving transformation $T$ on $\mathbf X$ is $\varepsilon$-almost mixing if for every $...
46
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4answers
5k views

Does anyone know an intuitive proof of the Birkhoff ergodic theorem?

For many standard, well-understood theorems the proofs have been streamlined to the point where you just need to understand the proof once and you remember the general idea forever. At this point I ...
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Rate of convergence of sojourn times of Markov chains

Let $(X_0,X_1,\dots)$ be a time-homogeneous Markov chain with finite state space $\Omega$. Assume that $(X_0,X_1,\dots)$ is irreducible and aperiodic and let $\pi$ be its stationary distribution. By ...
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2answers
168 views

Does ergodic theorem apply to trajectories outside of attractor?

Ergodic theorem says that $\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{t=1}^nf(T^tx) = \displaystyle\int f\,\mathrm{d}\mu$ for $\mu$-almost every $x$. In many cases, the support of $\mu$ ...
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1answer
120 views

Uniqueness of stationary measures for $(G,\mu)$ boundaries

Let $G$ be a countable group acting minimally by homeomorphisms on a compact Hausdorff space $X$ and $\mu$ be a probability measure on $G$ whose support generates $G$ as a semigroup. Let $\nu$ is a $\...
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1answer
406 views

Are these topological sequence entropy definition equivalent?

I am working on Möbius disjointness for models of topological dynamic systems. In that purpose, I try to understand the notion of topological entropy. We know, for a t.d.s $(X,T)$ that it is defined ...
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2k views

Alternating colors on a line: infinitely often or converge?

Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / …). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...
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Implications for a simple deterministic chaos definition

Among many others, one definition of deterministic chaos terms "chaotic" a classical dynamical system that satisfies the following three properties: sensitive dependence to initial ...
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98 views

Is there a condition for a subshift of finite type to be uniquely ergodic?

Are SFTs uniquely ergodic in general, or is there a known necessary and sufficient condition for them to be uniquely ergodic?
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Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?

It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure. Marc Peigné [1] constructed examples of geometrically infinite hyperbolic ...
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1answer
287 views

Can every ergodic map be approximated by ergodic maps close to the identity?

Let $\mathbf X := (X, \mathcal S, \mu)$ be a probability space without atoms. We say two measure preserving transformations $T$ and $F$ on $\mathbf X$ are $\delta$-close, for $\delta > 0$, if $ \...
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1answer
125 views

Uniformity of convergence in the pointwise ergodic theorem

Definitions and some motivation: Let $X$ be a compact metric space, and $T$ a uniquely ergodic measure preserving transformation on $X$, with associated invariant ergodic probability measure $\mu$. ...

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