Questions tagged [ergodic-theory]
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
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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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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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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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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 ...
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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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An example of an SRB measure which is not a physical measure
Let $f:M \to M$ be a $C^{1}$ diffeomorphism on a compact Riemannian manifold with a normalized Riemannian volume $\mathrm{Leb}$. Given an $f$-invariant Borel probability $\mu$ in $M$, we call the ...
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Are almost all measure-preserving flows on compact manifolds ergodic?
This may be a naive question, but I have been unable to find a reference that answers it directly, at least at a level that I can understand. My intuition from physics is that non-ergodicity is ...
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a mixing property on a tower
A tower $\Delta_g:=\{(x,n)\in X \times \{0,1,2,\cdots\}: n < R(x)\}$
where $R:X \to \{1,2,3,\cdots\}$ is a $L^1$ function on a probability space $(X,\mu)$, $g: X \to X$ is mixing and $\gcd \{R\}=1$,...
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Dynamics in the integers - Floor function
Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align*}
f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align*} where ...
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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
$$\|Aq-p\|^m < \frac{1}{T}, \|q\|^...
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Ergodic measures for the logistic map
$\DeclareMathOperator{\Inv}{Inv}\DeclareMathOperator{\Erg}{Erg}$This is mostly curiosity on my part and I hope that the MO community might be able to help.
For $c\in (0,4]$ consider the logistic ...
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Examples of different levels of the ergodic hierachy (specifically: weakly mixing & merely ergodic)
I am interested in generalizing some aspects of the ergodic hierarchy (of classical dynamical systems) to quantum theory. However, while I understand the definitions of the different levels of the ...
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Are $C^1$ vector fields generating an ergodic flow $C^0$ dense?
Note: This is a concrete case of the following question: Are almost all measure-preserving flows on compact manifolds ergodic?
Let $M$ be a Riemannian manifold with its natural Riemannian measure, and ...
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Metric entropy and topological entropy
It is well known that, for a dynamical system $T$ on a metric space $(X,d)$, the variational principle connects the definition of metric entropy and topological entropy. In other words,
if
$$M(X,T) := ...
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Partially hyperbolic systems and specification
Let $f: M \rightarrow M$ be a $C^{1+\alpha}$ diffeomorphism on a Riemannian compact manifold. Suppose that $f$ admits a dominated splitting $T M=E \oplus F$ with $E\ll F$, where $E$ is uniformly ...
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Probability of a finite cylinder set in a free group
Let $\mathbb{F}_n$ be the free group (each elemen is in its reduced form) generated by the set $\Sigma_n = \{a_1, a_2, \cdots, a_n, a_1^{-1}, a_2^{-1}, \cdots, a_n^{-1}\}$ and let $e$ denote the ...
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Zero entropy and the Koopman representation
Let $T$ be a measure preserving bijection of a probability space $(X,\nu)$. Consider the Koopman representation of $\mathbb{Z}$ on $L^2(X,\nu)$ given by $[z.f](x) = f(T^{-z}(x))$. The question is: can ...
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Absolute continuity of geometric sum of ergodic random variables
I have a stationary ergodic process of random variables $\{X_n\}_{n\in\mathbb{N}}$ and a constant $r<1$. Suppose that $\mathbb{E}(\log^+|X_1|)<\infty$. Then the sum
$$
\sum_{n=1}^{\infty} r^nX_n ...
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Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?
It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
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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 \bigcap_n ...
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Reference of the fact that Hoelder cocycles are associated to Hoelder potentials in Ledrappier's correspondence
Let $\tilde{M}$ be the universal cover of a compact pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary.
...
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Statistical characteristics of low complexity subshifts
I am looking for calculations of statistical characteristics (variance, entropy, etc.) of the $n$-dimensional distributions of the invariant measures of low complexity subshifts (e.g., the Sturmian or ...
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A density result for arithmetic progressions
Note: By upper/lower density, we shall mean the upper/lower asymptotic density as given here.
Question:
For any subset $S \subset \mathbb N$ with positive upper density, does there exists a $\...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...