Questions tagged [ergodic-theory]

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

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Reference request - logarithmic average

Consider a set $A\subseteq\mathbb{N}$. Consider an arithmetic function $a(n):\mathbb{N}\to\mathbb{C}$. I am looking for notation which describes the following:\begin{equation}\frac{\sum_{n\in A}\frac{...
alidixon222's user avatar
4 votes
2 answers
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Simple proof that exactness implies strong mixing

Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\...
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Continuous version of ergodic with integral

Let $f\in L^1(\mathbb{T}^n)$ such that $f \geq 0$ and $f$ might have a singularity, but along some curve or line $\xi(t): \mathbb{R}\to \mathbb{T}^n$ the value $f(\xi(t))$ is defined and continuous, ...
Sean's user avatar
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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}...
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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. ...
alidixon222's user avatar
2 votes
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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 $T$ ...
Scottish Questions's user avatar
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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. ...
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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 ...
Adam's user avatar
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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 $\...
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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 ...
Saúl Pilatowsky-Cameo's user avatar
6 votes
1 answer
278 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 ...
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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 ...
Cal's user avatar
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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 ...
Nikita Sidorov's user avatar
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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\...
Lau's user avatar
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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 ...
Kousaka_Reina's user avatar
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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 $R_{\Lambda}: I \to I$, ...
Alejo García Sassi's user avatar
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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 ...
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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 ...
Stepan Plyushkin's user avatar
8 votes
4 answers
758 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?
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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 ...
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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 ...
Stepan Plyushkin's user avatar
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2 answers
385 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 ...
Miranda's user avatar
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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{...
Dan's user avatar
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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 ...
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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 $(...
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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 ...
Ali Taghavi's user avatar
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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 \,...
Gabriel B. H. Lisboa's user avatar
7 votes
1 answer
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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 ...
Nate River's user avatar
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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 ...
homosapien's user avatar
3 votes
1 answer
292 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–...
user506835's user avatar
1 vote
0 answers
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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/...
User1723's user avatar
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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 $...
Matheus Manzatto's user avatar
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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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1 vote
0 answers
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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 ...
Julian Newman's user avatar
3 votes
1 answer
158 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, ...
Tiago's user avatar
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2 votes
1 answer
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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 ...
Augusto Santos's user avatar
3 votes
0 answers
126 views

the second largest eigenvalue of transfer operators

A Gauss map $T$ is mixing and satisfies Lasota-York inequalities. By Henon's theorem, we know that the transfer operator $\hat{T}$ associated with $T$ has a spectral gap. This means there exists a ...
user avatar
2 votes
0 answers
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Rotation number for multicomponent Schrödinger equation

Rotation number for Schrödinger equation of the form \begin{equation} -x''(t) +q(t) x(t) = E x(t) \end{equation} was defined in R. Johnson J. Moser "The rotation number for almost periodic ...
0x11111's user avatar
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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 ...
Ali Taghavi's user avatar
3 votes
1 answer
76 views

Spectral disjointness of unitary representations of Type I groups and orthogonality

Background: If $\mathcal{H}$ is a Hilbert space and $U:\mathcal{H}\rightarrow\mathcal{H}$ is a unitary operator, then for each $f \in \mathcal{H}$ the sequence $(\langle U^nf,f\rangle)_{n = 1}^\infty$ ...
Sohail Farhangi's user avatar
1 vote
1 answer
99 views

Nonamenable p.m.p. action on a standard probability space

Let $G$ be a discrete nonamenable countable group acting on a standard probability space $(X,\mu)$ through measure-preserving transformations. Is the action of $G$ always amenable? (Amenable action, ...
Ujan Chakraborty's user avatar
21 votes
1 answer
946 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 ...
Nate River's user avatar
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2 votes
0 answers
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Proof of Zimmer's cocycle super-rigidity theorem

I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...
John Depp's user avatar
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Proving light escapes mirrors via ergodic theory of billiards

There's a longstanding open problem concerning whether or not it's possible to trap all the light from a point source using a finite collection of circles/lines whose sides are mirrors. This seems ...
interstice's user avatar
1 vote
0 answers
112 views

Mathematical justification for the use of an energy shell in the microcanonical ensemble

I would like to understand an identity used in the deduction of the explicit formula for the probability distribution of the microcanonical ensemble in statistical mechanics. Consider $\Lambda$ to be ...
MathMath's user avatar
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3 votes
1 answer
226 views

"Ergodic theorem" for Markov kernels

Consider a discrete time Markov chain $(X_t)$ on a finite state space $\mathcal{S}$, with transition matrix $P$. Assume that the chain admits a stationary distribution $\pi$, which I will identify ...
Francesco Bilotta's user avatar
1 vote
0 answers
91 views

Convex combination of positive mean-ergodic operators

Let $T_1,T_2:L^1([0,1],\mathrm{d}x)\to L^1([0,1],\mathrm{d}x)$ be positive mean-ergodic operators such that: For every $h:[0,1]\to \mathbb{R}_+$ we have that $$\int_0^1 T_1 h(x)\mathrm{d}x = \int_0^1 ...
Matheus Manzatto's user avatar
2 votes
0 answers
233 views

Correlation decay rate

Let $T$ be a continuous transformation of a probability measure space $(X,\mathcal{B}(X),\mu)$ and $\varphi ,\phi \in L^2(\mu)$ (so-called observable) . The correlation function of $\varphi ,\phi$ (a ...
Mrcrg's user avatar
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1 answer
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Example of finite closed cover with entropy strictly greater than topological entropy

I'm reading "Topological entropy bounds measure-theorettic entropy", by L.W. Goodwyn. enter link description here After Proposition 2, he mentions that "finite closed cover can yield ...
felcove's user avatar
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3 votes
1 answer
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Competing definitions of smooth orbit equivalence relation

Suppose that $X$ is a standard Borel space (meaning it is endowed with a $\sigma$-algebra coming from some Polish topology on $X$) and $G$ is a Polish group acting in a Borel way on $X$. Denote by $...
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