# Questions tagged [ergodic-theory]

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

730
questions

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### 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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129 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 $...

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149 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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98 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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52 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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249 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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60 views

+100

### 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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43 views

### 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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72 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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81 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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90 views

### 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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104 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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254 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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100 views

### Metropolis-Hastings sampling as a group action

Suppose that you have a topological space $\Omega \subset \mathbb R^n$ accompanied a measure $\mu$ and you're running an iterative sampling algorithm like Metropolis-Hastings. To sample you choose a ...

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57 views

### convergence rate for ergodic Markov chains induced by stable dynamical systems

Consider a deterministic dynamical system on $\mathbb{R}^n$ defined by the recurrence $x_{t+1} = f(x_t)$.
Suppose the dynamical system is stable in the following sense: there exists a $Q : \mathbb{R}^...

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65 views

### Random walks on the Poincaré disk

Let $G$ be the group of isometries of the Poincaré disk. Let $\mu$ be a probability measure on $G$, and consider $g_1,..,g_n$ i.i.d. random variables on $G$ distributed according to $\mu$. For $z\in \...

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59 views

### Why measure hyperfinite is equivalent to hyperfinite except for a compressible set?

According to Kechris' paper "The theory of countable Borel equivalence relations" (pp.82), a countable Borel equivalence relation E is measure hyperfinite iff there is an E-invariant Borel ...

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77 views

### When the Jacobian of unstable measure converges

Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...

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120 views

### An angle between two vectors in Oseledets theorem

Let $f:\Sigma \to \Sigma$ be a two side shift map, where $\Sigma=\{1,2,3,4\}^{\mathbb{Z}}$ and let $A:\Sigma \to SL(2,\mathbb{R})$ be a function such that $A((x_{n}))=A_{x_{0}}$. Assume that there are ...

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54 views

### Number of permitted words up to permutation in a subshift

Let $A$ be a finite set and let $X \subseteq A^{\mathbb{N}}$ be a subshift. Let $\mathcal{L}_n$ denote the set of words of length $n$ appearing in $X$. For a word $w \in \mathcal{L}_n$, one can ...

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25 views

### Nonintegrable classical dynamical systems and deterministic chaos

I'm trying to delineate a minimal (and informal) "taxonomy" for classical continuous dynamical systems that could be interested by the phenomenon of "chaos" - unfortunately the ...

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52 views

### When is the unstable direction map $x\mapsto e^{u}(x)$ injective?

Let $f:M \to M$ be a $C^{2}$-Anosov diffeomorphism. Therefore, there exists an invariant splitting of the tangent bundle $T_{x}M = E^s(x) \oplus E^u(x)$ into a stable and an unstable directions, that ...

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136 views

### Theoretical invariant distribution of discrete dynamical systems, including the Riemann Zeta map

Update on 3/10/2021: I added Example 5 in the Appendix. This generic example encompasses the Riemann Zeta dynamical system. A simple version of this post, targeted to engineers, machine learning ...

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85 views

### Values appearing with density in an ergodic system

Values appearing with density in an ergodic system
Let $(X,\mu)$ be a probability space with invertible, measure preserving, totally-ergodic map $T:X \to X$. ($(X,\mu,T)$ is a $\mathbb{Z}$ dynamical ...

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128 views

### Are topologically free and essentially free equivalent for minimal spaces with invariant measures?

Suppose $G$ is a discrete group acting by homeomorphisms on a compact Hausdorff space $X$, such that the action is minimal. Fix an invariant Radon measure $\nu$ on $X$. Is topologically free (the ...

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85 views

### Is it true that $(X,T^k)$ minimal for all $k\geq1$ implies $\mathrm{Aut}(X,T) = \mathrm{Aut}(X,T^k)$ for all $k\geq1$?

Let $(X,T)$ be a topological dynamical system ($X$ is compact metric space and $T\colon X\to X$ a homeomorphism). Recall that its automorphism group is
$$ \mathrm{Aut}(X,T) = \{g\colon X\to X : \text{$...

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101 views

### Example of topologically transitive dynamical system with invariant non-ergodic Borel measure

Let $U \subset M$ be an open subset of a Riemannian manifold. I’m trying to find or construct an example of a topologically transitive dynamical system $f : U \to U$ for which
$f : \Lambda \to \...

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54 views

### Disjointness of Bernoulli shifts and zero entropy systems

I am looking for a reference for a proof of the following fact: if $(X, \mu, T)$ is a measure-preserving system isomorphic to a Bernoulli shift and $(Y, \nu, S)$ has zero (Kolmogorov-Sinai) entropy, ...

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145 views

### Two generalizations of the Verblunsky Theorem

I learned from this paper about the Verblunsky theorem.
My question is that: What kind of generalizations of this theorem is availlable?
In particular I am interested in the following two possible ...

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334 views

### Laplacian on manifolds and random matrix theory

Let $M$ be a compact Riemannian manifold with a metric $g$, and consider the spectrum of the Laplacian operator $\Delta$.
What is known about the relationship between this spectrum and random matrix ...

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99 views

### Pocket billiards with balls in general position

There were at least two earlier MO questions about ideal pocket billiards.
(Ideal: frictionless, perfectly elastic collisions.)
Perfectly centered break of a perfectly aligned pool ball rack.
Does ...

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328 views

### Ergodicity of the action of $\operatorname{SL}(n,\mathbb R)$ on $\operatorname{SL}(n,\mathbb R)/\operatorname{SL}(n,\mathbb Z)$

$\DeclareMathOperator\SL{SL}$Let $G\mathrel{:=}\SL(n,\mathbb R)$ and $\Gamma\mathrel{:=}\SL(n,\mathbb Z)$. Consider the action of $G$ on $(G/\Gamma,\mu)$ by left translation, where $\mu$ is the Borel ...

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55 views

### Examples of minimal almost 1-to-1 extension of torus having positive entropy?

It is well known that Toeplitz subshifts are minimal almost 1-to-1 extensions of an odometer, and that some of these subshifts have positive entropy. Thus, even if a system is an almost 1-to-1 ...

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66 views

### Is there a term for a linear operator on an $L^p$ space that “locally respects boundedness”?

Let $X$ be a Polish space, and $\mu$ a locally finite measure. Take any $p \in \{0\} \cup [1,\infty)$. We will say that a linear operator $T \colon L^p(\mu) \to L^p(\mu)$ has property $(\ast)$ if ...

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60 views

### Recurrence results for an “on average” measure preserving transformation

I have a finite measure space $(X, \mathcal{S}, \mu)$, and a transformation $f:X\rightarrow X$ that "preserves measure on average". That is, for $A \in \mathcal{S}$
$$
\lim_{n\rightarrow \...

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99 views

### Sets of invariant measures of Markov operators

A family of Markov operators $P_i \colon C \to C, i \in I$ is given. Let $V_i$ be the set of the $P_i$-invariant measures. Is there any result in the literature about a necessary and sufficient ...

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95 views

### Uniform convergence for pointwise ergodic theorem

Let $K$ be a compact set of $\mathbb{R}^n$ $(n\geq 1)$ and $v\in\mathcal{C}^1(K,\mathbb{R}^n)$ be a speed field on $K$ such that for any initial condition $x_0\in K$, the following dynamical system
\...

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376 views

### Sequences similar to $\{n\alpha\}$ that are both equidistributed and truly random-like

See update at the bottom.
Here the brackets represent the fractional part, and $\alpha \in [0, 1]$ is a positive irrational number. It is well known that the sequences $\{n\alpha\}$, $\{n^2\alpha\}$ ...

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59 views

### Discrepancy estimate for $3$-interval exchange or $n$-interval exchange map, $n\geq 3$

We know that 2-interval exchange on $\mathbb{T}$($\mathbb{T}$ is identified with $[0,1]$ for convenient in the follow context) is just a rotation on $\mathbb{T}$, and there is a process called ...

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468 views

### Are there numbers whose binary and ternary representations simultaneously have few digit transitions? How frequent are those numbers?

For a natural number $n$, let $c_b(n)$ denote the number of digit transitions in the representation of $n$ in base $b$. By a digit transition, we mean a pair of successive, unequal digits: for ...

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21 views

### Proof of property for Fiedland entropy

I am working with Friedland entropy and there is a proof I cannot figure out how to do.
Friedland entropy is defined for $\mathbb{Z}^k$ continuos actions $\mathcal{T}$ on a topological metric space $X$...

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142 views

### Is Birkhoff's ergodic theorem true for $L_\infty$?

Is Birkhoff's pointwise/individual ergodic theorem for $L_\infty.$ Clearly, it is true if the measure space is finite? What about the measure space not finite?

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82 views

### Lyapunov spectrum($h_{\mathrm{top}}(K(\alpha))$) achieves a positive value somewhere

$\DeclareMathOperator{\top}{\mathrm{top}}$Let $(\Sigma, T)$ be a topologically mixing subshift of finite type and $f:\Sigma \to \mathbb{R}$ be a Hölder continuous map.
Let
$$K(\alpha)=\Big\{x\in \...

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114 views

### Detecting isolated eigenvalues from local spectral measures

Please note: This question has been edited after it became clear from Christian Remling's answer that the original formulation was far from what I really meant to ask.
Let $T\ne 0$ be a self-adjoint ...

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66 views

### Distribution of the values of the product $\prod_{k=1}^n |1-e(k\alpha)|$ for an irrational number $\alpha$

For an irrational number $\alpha$, let $e(k\alpha):=\exp(2k\pi i\alpha)$. It was indicated in this thread that
$$\limsup_{n \to \infty} \prod_{k=1}^n |1-e(k\alpha)|=\infty$$
(actually a weaker result ...

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128 views

### Vandermonde shift

I'm looking for any known results on a shift operator commutated by a Vandermonde matrix. That is, let
$$T=\begin{bmatrix}0 & 1 & 0 & 0 & \cdots \\
0 & 0 & 1 & 0 & \...

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149 views

### finiteness of moments of the stationary distribution of a Markov chain

I have a Markov chain $\{X_k\}_{k\geq 0}$ on $\mathbb{R}$. The corresponding probability density functions satisfy
$$
f_{k+1}(t) = \int_{-\infty}^\infty \Psi(t,\tau)f_k(\tau)\,d\tau,\qquad k=0,1,2,\...

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81 views

### Rotation set vs existence of rotation number

Let $f\colon \mathbb{S}^{1}\to\mathbb{S}^{1}$ be a continuous function of degree 1 and $F\colon \mathbb{R}\to \mathbb{R}$ a lift of $f.$ One can define, for each $x\in \mathbb{R}$, the rotation number ...

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208 views

### Does such a function exist?

I am looking for a function with the following property:
Let $v_1,v_2$ be two linearly independent vectors in $\mathbb{R}^2.$
I am given a smooth function $g:(0,1) \rightarrow (0,\infty).$
I am trying ...

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85 views

### Ergodicity of a dynamical system on the $n$-sphere

Let $v$ be continuous and nowhere-vanishing vector field tangent to the $n$-sphere $\mathbb{S}^n$ (hence $n$ is odd, w.r.t the Hairy-Ball Theorem). Let $x$ be a trajectory on $\mathbb{S}^n$, defined ...