# Questions tagged [ergodic-theory]

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

876
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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 ...

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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 ...

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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\...

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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 ...

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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 $R_{\Lambda}: I \to I$, ...

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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 ...

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### When does the Shannon-McMillan-Breiman Theorem fail?

Theorem (Shannon-McMillan-Breiman): Let $(X,\mathscr{B},\mu,T)$ be an ergodic measure preserving system on a Borel probability space, and let $\xi$ be a countable partition with finite entropy. Then,
$...

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### 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 ...

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### 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 ...

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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{...

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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 ...

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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 \,...

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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 ...

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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 ...

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### 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–...

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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/...

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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 $...

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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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### 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 ...

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### 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, ...

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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 ...

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### 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 ...

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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 ...

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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 ...

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### 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 $(\langle U^nf,f\rangle)_{n = 1}^\infty$ ...

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### 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, ...

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### 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 ...

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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. ...

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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 ...

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### 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 ...

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### "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 ...

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### 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 ...

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### 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 ...

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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 ...

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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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### Persistence of KAM tori as a function of dimension

I have tried posting this question in MSE, but I think it might be too technical so I'm trying again here.
In KAM theory one tries to describe the persistence of quasi-periodic motion when an ...

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### The definition of simple eigenvalue

This question was posted a long time ago on the mathexchange, but I didn't get any answers there, and despite having discussed it with some colleagues, I don't think I have a definitive answer.
I am ...

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### 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 $X$, ...

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### A sensitive 2-dimensional cellular automaton with a blocking word

I'am a Ph.D student in the domain of discrete dynamical systems. My thesis is about spectral properties of cellular automata in higher dimension.
Kurka gives a classification for one dimensional ...

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### The liminf of an expression involving an irrational rotation

Let $0 < a < 1$ be an irrational number. Is it true that
$$\liminf_{n \in \mathbb N, n \to \infty} n \{na\} = 0?$$
Note: Here $\{\cdot\}$ denotes the fractional part.

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### A uniform distribution problem coming from higher dimensions

Thinking about an approximation problem related to random walks, the following question came up.
Suppose we have $m$ numbers $a_1, \ldots, a_m \in \mathbb{R}$ and that $b \in \mathbb{R}$ is not in the ...

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### Equivalence of the definitions of exactness and mixing

Let $f:X \to X$ be a continuous map, where $X$ is a compact metric space. We say that $f$ is (locally) expanding if there are constants $\lambda >1$ and $\delta_0 > 0$ such that, for all $x, y\...

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### Difference between the topological entropy and Hausdorff dimension for multifractal formalism

I have been reading some results about multifractal formalism. I noticed that some results were proved for the Hausdorff dimension and some results for the topological entropy (in the sense of Bowen). ...

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### Existence of a domain with simple Dirichlet eigenvalues

Let $g$ be a smooth Riemannian metric on $\mathbb R^3$ that coincides with the Euclidean metric outside a compact set $K$. Does there exist some domain $\Omega$ with smooth boundary such that $K \...

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### Two ergodic decomposition theorems

Let $X$ be a compact metric space and $T\colon X \to X$ a continuous map. Additionally, let $\mathcal{M}^T(X)$ be the set of $T$-invariant probability measures on $X$ and $\mathcal{E}^T(X)$ the set of ...