Questions tagged [ergodic-theory]

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

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5 votes
1 answer
376 views

Positiveness of the largest Lyapunov exponent

Let $\alpha\in \mathbb{R} / \mathbb{Q}$, let $A(x)$ be the $2$-by-$2$ matrix $$ A(x)=\begin{pmatrix} \dfrac{1}{{\lambda}^2}-2 \cos 2\pi x -1& 2\lambda \cos 2\pi x-\dfrac{1}{{\lambda}} \\ \dfrac{...
6 votes
2 answers
556 views

If the average of a sequence converges, can I find a uniform bound that does not depend on where I start?

Let $\{a_k\}_{k\in \mathbb{Z}} \subset \mathbb{R}$ a real sequence and $a\in \mathbb{R}$ such that $$ \lim_{n\to +\infty} \frac{1}{n} \sum_{k=1}^n a_k = a = \lim_{n\to +\infty} \frac{1}{n+1} \sum_{k=0}...
2 votes
1 answer
290 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 $T$, ...
2 votes
2 answers
302 views

Existence of the limit of periodic measures

Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
4 votes
2 answers
422 views

Reference request - Pillai-Selberg Theorem

I want to find a proof of the following claim. Let $\Omega(n)$ denote the number of prime factors of an integer $n$ counted with multiplicity. Then $\Omega(n)$ equidistributes over residue classes. ...
3 votes
1 answer
291 views

Equidistribution of the orbit $\{\text{diag}(t^a,t^{-a})\Lambda \}_{t>0}$ for a.e. $\Lambda\in \text{SL}(2,\mathbb R)/\text{SL}(2,\mathbb Z)$

$\DeclareMathOperator\diag{diag}\DeclareMathOperator\SL{SL}$It is well-known that geodesic flow $g_t=\{\diag(e^t,e^{-t}) \}_{t>0}$ acts ergodically (actually mixing) on $\SL(2,\mathbb R)$ (Howe–...
2 votes
1 answer
111 views

Koopman operators on $L^p(X)$

On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ ...
14 votes
7 answers
3k views

Furstenberg $\times 2 \times 3$ conjecture, bibliography

Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure. I wanted to have a ...
1 vote
0 answers
85 views

Mixing for a gas of hard spheres

The gas of hard spheres is a model for a gas in a container, where each particle is a sphere of radius $\epsilon$. The spheres interact with each other and with the container with elastic collisions. ...
12 votes
6 answers
1k views

Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$

Consider the sequence in the unit disk $D=\{(x,y)\,|\,x^2+y^2\leq 1\}$ iteratively defined by the quadratic map $$\begin{aligned} x_{n+1}&=2x_ny_n\\y_{n+1}&=1-2x_n^2\end{aligned},$$ starting ...
3 votes
1 answer
157 views

Quantitative version of ergodic theorem in Markov chains

Consider an irreducible Markov chain $X_t$ with finite state space $E$, and unique invariant measure $\pi$. Fix a function $V:E\to\mathbb R$ such that $E_\pi[V]=0$. The ergodic theorem tells us that, ...
15 votes
1 answer
1k views

Krein Milman theorem without the axiom of choice

The Krein-Milman theorem asserts that in a locally convex topological vector space, a nonvoid compact convex subset is the closed convex envelope of its extreme points. But I would like to know when ...
6 votes
1 answer
276 views

Sequence of digits of powers of two

Elementary number theory tells us a lot about the final digits of the powers of two, and ergodic theory (more specifically the theory of equidistribution of points in the orbit of an irrational ...
2 votes
2 answers
246 views

If $\inf\{b\in\mathbb{R}\mid\sum_{n=1}^{\infty}e^{-ax_n-by_n}<+\infty\}=1-a$ for all $a\in [0,1]$, does this equality hold for all $a\in\mathbb{R}$?

Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\...
11 votes
2 answers
2k views

De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales

De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\infty$ that is ...
3 votes
1 answer
365 views

Birkhoff ergodic theorem for ergodic Markov processes

This question was previously posted on MSE. This question might be easy but I am really stuck on it. Let $M$ be compact metric space and $\mathcal B(M)$ the Borel $\sigma$-algebra of M. Consider the ...
12 votes
3 answers
878 views

Looking for at least one beautiful and not too technical result in asymptotic group theory

We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this ...
0 votes
1 answer
246 views

Do invariant open sets generate the $\sigma$-algebra of invariant sets?

Let $X$ be a Polish space with Borel $\sigma$-algebra $B(X)$. Let $G$ be a locally compact group. $T:G\times X\to X$ be a continuous action of $G$ on $X$. The $\sigma$-algebra of invariant sets is ...
-5 votes
1 answer
558 views

Central limit theorem for irrational rotations

Let $\alpha$ be an algebraic integer of modulus 1, and $ R_\alpha z=\alpha z$. Is $$\lim_{n\to\infty}\frac{\log|\sum_{k=1}^n \Re R_\alpha^k z|}{\log n}=\frac12$$ for all $z\in S^1$? Birkhoff's ergodic ...
2 votes
1 answer
218 views

Does Bernoulli imply exponential mixing?

This question comes from this paper where the authors proved that exponential mixing implies Bernoulli. They also mentioned in the introduction that Bernoulli is the strongest ergodic property and ...
4 votes
1 answer
197 views

Ergodic actions and deviation from invariance

Let $M$ be a von Neumann algebra and let $(\phi_t)$ be an ergodic point-$\sigma$-weakly continuous one-parameter group of automorphisms $\phi_t\in \mathrm{Aut}(M)$, i.e., $\Vert\omega-\omega\circ\...
1 vote
1 answer
67 views

Number of ergodic transverse measures for geodesic laminations - bounded by the genus?

Consider a geodesic lamination $\Lambda$`of a closed hyperbolic surface $S$ of genus $g$, and take a globally transverse closed curve $I$. The lamination induces a return map $R_{\Lambda}: I \to I$, ...
1 vote
0 answers
94 views

Question about ergodic flows and periodicity

Let $X$ be a compact Haussdorf space, let $\mu$ be a Borel measure on $X$ with $\mathrm{supp}(\mu)=X$ and let $(\phi_s)_{s\in\mathbb R}$ be a one-parameter group of homeomorphisms which is continuous ...
1 vote
0 answers
264 views

Using the von Neumann crossed product to introduce a measure on the orbit space?

Suppose we're given an action (possibly: ergodic) of a group G (say, $\mathbb{R}$) on a measure space $(X, \mu)$ (possibly: a standard probability space). Question: is there a natural way of using the ...
21 votes
1 answer
920 views

Roadmap to Ergodic Theory

I have recently been interested in going deeper into ergodic theory, beyond an introductory level of knowledge. Background wise, my training has mostly been in stochastic analysis, and I have a ...
8 votes
4 answers
743 views

Ergodic theory applied to number theory

I am interested in the links between Ergodic Theory and Number Theory. Can anyone give some references for papers to read in this field? Any open problems? Or ideas where it may be applicable in NT?
7 votes
2 answers
382 views

Let $(a_n)_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the set$\{2^n3^m\mid m,n\in N\}$. Find $α$ such that $(a_n)\alpha\pmod1$ is not equidistributed

Let $$(a_n)_{n \in \mathbb{N}} = (1,2,3,4,6,8,9,12,16,18,\cdots)$$ be a sequence that is a listing of the set $$\{2^n3^m \mid m,n \in \mathbb{N}\}$$ We need to find an irrational number $\alpha$ such ...
3 votes
1 answer
70 views

Rate of convergence for Markov chain in random environment

Let $(\Omega,\mathfrak{F},\mathbb{P})$ be a probability space and $\sigma:\Omega\to\Omega$ be an ergodic, invertible and measure preserving transformation. Consider a family of column stochastic ...
1 vote
0 answers
195 views

Are orbits of a measurable flow always measurable with measure zero?

Let $(X, \mathcal{B})$ be a standard Borel space with a probability measure $\mu$ on $\mathcal{B}$. Let $(T_t)_{t \in \mathbb{R}}$ be a jointly measurable flow (i.e. $(T_t)_{t \in \mathbb{R}}$ is a ...
1 vote
0 answers
263 views

Intersection of thickly syndetic sets

Question: Let $\Gamma$ be a countable group. Is the intersection of two thickly syndetic sets still thickly syndetic? I've only seen the proof for the group $\mathbb{Z}$ (and I believe this method ...
0 votes
1 answer
171 views

Invariance of the Kronecker factor

Let $(X,\mathcal{F},\mu,T)$ be a measure preserving system and $U_T$ Koopman operator on $L^2(X)$, i.e. $U_T f = f\circ T$. Note that, for the moment, I am not imposing any further assumptions on $X$, ...
3 votes
0 answers
144 views

General references on dynamics of continuous, piecewise linear interval maps

I want to find a general reference on topological and measurable dynamics of continuous piecewise linear interval maps. I am particularly interested in cases with only three pieces. Even ...
0 votes
1 answer
98 views

If $a_1=1$ and $a_n=\sec (a_{n-1})$ then what does the proportion of positive terms approach, as $n\to\infty$?

Consider the sequence $a_1=1$ and $a_n=\sec (a_{n-1})$ for $n>1$. What does the proportion of positive terms approach, as $n\to\infty$? At first I thought the limiting proportion might be $\frac{...
3 votes
0 answers
213 views

Ergodic Markov operator

Given a $\sigma$-additive measure space $(E,\Sigma,\mu)$. A Markov operator $P : L^1(\mu) \to L^1(\mu)$ is a linear operator with $ f \geq 0 \Rightarrow Pf \geq 0 $ $ f \geq 0 \Rightarrow \|Pf\| = \|...
2 votes
0 answers
317 views

A (possible) generic spectral property in one dimensional dynamics

Context and Definitions Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [1]) if: $T$ has a finite number of hyperbolic periodic attractors; and defining $...
1 vote
1 answer
249 views

Alternate definitions of compact and weak mixing extensions

In Furstenberg's proof of the multiple recurrence theorem in ergodic theory, one makes use of the concept of compact and weak mixing extensions of a measure preserving system. The following definition ...
2 votes
1 answer
134 views

Ergodicity in irrational polygons and rational polyhedra

Steven Kerckhoff, Howard Masur and John Smillie in their paper [1] have proved that on rational polygons "For almost every direction the geodesic flow/billiard flow is ergodic". Is there any ...
7 votes
5 answers
765 views

orbits in locally compact group

As everyone knows if $x\in S^1$, then the set $\{ x^n \}$ is either finite or dense. Under which condition is true for any other locally compact group, i.e if $G$ is a locally compact group, and $x\in ...
-1 votes
1 answer
132 views

Strong law of large numbers for a sequence of random variables in different probability spaces

Is it known whether the following version of the strong law of large numbers holds? For each $k\in\mathbb{N}$, let $\Omega_k$ be a finite set and $\mu_k$ be a probability measure on $\Omega_k$. Let $(...
1 vote
0 answers
146 views

The space of ergodic elements of a topological or Lie group

Let $G$ be a compact topological group with normalized Haar measure $\mu$. An element $g\in G$ is an ergodic element if the mapping $L_g:G \to G $ with $x\mapsto gx$ is an ergodic map. The ...
0 votes
0 answers
54 views

Distortion lemma for composition of (distinct) functions expanding on average

I am trying to describe the following dynamics: Let $(T_{\rho})_{\rho \in [0, 1]}$, $T_{\rho}: [-1, 1] \rightarrow [-1, 1]$ be a family map which satisfies: $\forall \, \rho \in [0, 1], \, \exists \,...
7 votes
1 answer
995 views

If the pointwise ergodic theorem holds along all subsequences with nonzero natural density, is the system strong 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 whose natural density exists ...
1 vote
0 answers
53 views

Expected value for minimum denominator of arbitrarily chosed rational out of a ball of fixed radius to complex plane

So I have a research problem which states that we compute the probability mass function of the random variable which returns the smallest denominator of a reduced fraction in a randomly chosen real ...
2 votes
0 answers
248 views

When is $f^*:T^*M\to T^*M$ an ergodic map for a diffeomorphism $f:M\to M$?

Let M be a differentiable manifold and $f:M \to M$ be a diffeomorphism. Then $f$ induces a natural map $f^* :T^*M \to T^*M$. The pull back map $f^*$ is a symplectomorphism wrt the ...
0 votes
1 answer
405 views

Recurrence and transience of cocycle over a dynamical system

Let $X$ be a compact metric space, $T$ a homeomorphism on $X$ and $\mu$ a $T$-invariant probability measure. Let $\phi:X\to\mathbb{R}$ be a continuous function and $\phi_n(x)=\phi(x)+\cdots+\phi(T^{n-...
16 votes
2 answers
2k views

Random walk is to diffusion as self-avoiding random walk is to ...?

One can view a random walk as a discrete process whose continuous analog is diffusion. For example, discretizing the heat diffusion equation (in both time and space) leads to random walks. Is there a ...
2 votes
1 answer
236 views

Ergodicity of linear dynamical systems and convergence of covariance matrices

Let $z(n+1)=Bz(n)+\xi(n+1)$ be an $N$-dimensional linear dynamical system with $\left(\xi(n)\right)_{n\in\mathbb{N}}$ being i.i.d. with $\xi(n)\sim\mathcal{N}(0,\Sigma_{\xi})$. Assumptions: a) The ...
1 vote
0 answers
73 views

Understanding logarithmic law for geodesics

I was reading this seminal paper https://projecteuclid.org/journals/acta-mathematica/volume-149/issue-none/Disjoint-spheres-approximation-by-imaginary-quadratic-numbers-and-the-logarithm/10.1007/...
1 vote
0 answers
62 views

Is it known whether 2-mixing continuous systems on a compact metric space are necessarily "pseudo-3-mixing"?

I asked this question on Math Stack Exchange at https://math.stackexchange.com/questions/4739742/; it received 4 upvotes, but no comments or answers even after a 450-point bounty. The question: Is ...
0 votes
0 answers
128 views

Ruelle's Theorem

When reading the spectral theorem of Ruelle's operator, a crucial question arises: Is there an effective method to compute the leading eigenvalue and the equilibrium state explicitly? Let's consider a ...

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