Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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

0
votes
0answers
46 views

size of local strong stable manifold is measurable

Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$. There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...
0
votes
0answers
52 views

Stone Space in Ergodic Theory

Any use of Stone space in Ergodic theory? Where I mean for example replace in theorems where you assume Polish space or any other suitable spaces which appear in classical ergodic theory literature, ...
1
vote
0answers
68 views

Supremum over all invariant Borel probability measures of the ergodic averages ratio of rates

Let $M$ a two-dimensional compact manifold and $f:M\to M$ a diffeomorphism $C^r$, $r\geq 2$ and $f(x,y)=(mx,\lambda y)$ where $m:M\to \mathbb{R}$ and $\lambda:M\to \mathbb{R}$ ,$\lambda<1<m$. ...
3
votes
0answers
159 views

Fundamental group and group measure space construction

Let $N$ be a type ${\rm II}$ factor, with trace $\tau$. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq ...
12
votes
0answers
157 views

Statistics for rational points on curves of genus $g$ over $\mathbb{F}_q$, $g\gg q$

Consider the distribution of the number of $\mathbb{F}_q$ points as I range over smooth projective curves of genus $g$ (defined over $\mathbb{F}_q$). If $q\gg g,$ the Hasse-Weil bounds give me a lot ...
2
votes
0answers
42 views

Sufficient condition for square root fluctuations of an ergodic sequence

Suppose I have a random sequence $\mathbf{X}=\{X_n\}_{n\in\mathbb{Z}}\subset \mathbb{R}^{\mathbb{Z}}$ that is ergodic with respect to translations. I am interested in a sufficient condition on $\...
2
votes
0answers
127 views

Anzai flow in noncommutative geometry

Consider Anzai flows (cf. Anzai: Ergodic Skew Product Transformations on the Torus, Osaka Math. J. 3 (1951), 83-99) on the two dimensional torus $T^2$. I would like to know if there exists some ...
3
votes
0answers
78 views

Counting lattice points in adelic spaces

Let $\mathbb{A}$ denote the ring of adeles of $\mathbb{Q}$, let $\mu$ be the Haar measure of $\mathbb{A}$, and let $\|\cdot\|_{\infty}$ denote the sup-norm of the components in the Archimedean of $\...
5
votes
1answer
191 views

Cartan subalgebra and group measure space construction

Let $N$ be a ${\rm II}_1$ factor. A maximal abelian self-adjoint subalgebra (MASA) is a $*$-subalgebra $A \subset N$ such that $A' \cap N = A$. It is called a Cartan subalgebra if moreover $\mathcal{N}...
0
votes
1answer
71 views

Irrational natural density set, intersected with odd polynomial

Let $A$ be a set of integers with irrational natural density. That is, suppose that $\lim_{n\to\infty}\frac{\#(A\cap [-n,n))}{2n}$ exists and is irrational. Denote this value by $\alpha$. Now let $p$...
6
votes
1answer
118 views

Density-$c_0$ in $\ell^\infty$

Let $A \subseteq \mathbb{N}$, define the upper density of $A$ as, $$ \overline{\delta}(A) := \limsup_{N\to\infty}\frac{|A\cap\{1,2,3,\cdots,N\}|}{N}. $$ This naturally leads to a weaker form of ...
2
votes
1answer
73 views

stationary measure for linear cocycle(random transformation matrices)

Let $(M,\mathcal B, \mu)$ be a probability space which $M=\{A_{1},A_{2},...,A_{N}\}^{\mathbb{N}}$ ($A_{i} \in GL(d ,\mathbb{R})$) and $\mu=p^{\mathbb{N}}$. Let $F:M\times \mathbb R^d\to M\times \...
3
votes
1answer
99 views

A question about distribution of fractional part of $2^k\alpha$

Let $\{x\}$ be the fractional part of $x$, i.e. $\{x\}=x-[x]$, where $[x]$ is the biggest integer $\leq x$. The question might be well known but I don't know where to look for: Assume $\alpha$ is an ...
10
votes
2answers
286 views

Minimal, uniquely ergodic but not Lebesgue-ergodic?

So here's my question: Does there exist a minimal diffeomorphism of class at least $\mathcal{C^2}$ of a compact manifold X which is minimal uniquely ergodic with unique probability measure $\mu$ ...
2
votes
0answers
104 views

Renyi's theorem on mixing

I have been trying to understand the proof of Renyi's characterization of (strongly) mixing transformations: A measure preserving transformation $T \text{ is strongly mixing iff for every measurable }...
13
votes
3answers
625 views

Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?

Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$. Does anyone know of an ...
1
vote
1answer
62 views

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 function in a complete metric ...
1
vote
0answers
28 views

Strong ergodicity of a countable subgroup of $PO(3,1)$

If we identify the boundary at infinity of the hyperbolic $3$-space $\mathbb{H}^3$ with the complex projective line $\mathbb{P}^1(\mathbb{C})=\mathbb{C} \cup \{ \infty\}$, we know that the ideal ...
2
votes
1answer
94 views

time delay ergodic theorem

given dynamic system $(X, \mathcal{B}, F, \mu), \mu \circ F^{-1}=\mu, F $ is mixing, $ A \in \mathcal{B}, s.t. \mu(A) >0 $. consider dynamic system $(X\times X, \mathcal{B}\otimes \mathcal{B}, ...
1
vote
0answers
48 views

Equivalent condition for Poincare polynomial

I have found a statement in the introduction of the paper 'Sets of Recurrence and Generalized Polynomials' by Bergelson & Haland, which is Result: Given a polynomial $p \in \mathbb{R}[x]$ such ...
3
votes
0answers
94 views

Maximal ergodic theorem on some dyadic intervals

What we refer to maximal ergodic theorem in this thread is the following: let $\left(\Omega,\mathcal F,\mu\right)$ be a probability space and let $T\colon\Omega\to \Omega$ be a measurable and measure ...
2
votes
0answers
86 views

Generalized right Perron-Frobenius eigenvector with rationally independent coordinates

Suppose you are given a directed graph $G=(V,E)$ which is strongly connected, i.e. for every two vertices $u,v \in V$ there exists a directed path between them. Consider the corresponding edge shift ...
1
vote
0answers
72 views

Computing algebraic entropy

Could you recommend any reference for computing algebraic entropy? Here algebraic entropy is defiened as $\lim_{n \to \infty}\log (deg (f^n))^{1/n}$ for a rational map $f $. I saw that there are ...
3
votes
0answers
116 views

Identification of ultrafilters with measures

We know that each ultrafilter $p$ on $\mathbb{N}$ can be identified with a finitely additive $\{0,1\}$-valued probablity measure $\mu_{p}$ on the power set of $\mathbb{N}$. Now my question is which ...
3
votes
0answers
125 views

Random $\beta$-transformation and its limit theorem

given probability space $ (\Omega, T, \mu), \mu$ is ergodic and $ T $ is invertible ( can regard $T$ as two sides shift) define random $\beta$-transformations: random variable $\beta:\Omega \to (1,\...
4
votes
2answers
231 views

A kind of converse to the Hopf theorem on ergodicity of geodesic flow in negative curvature

Is there a 2 dimensional Riemannian manifold $M$ whose curvature is not negative but its geodesic flow is an ergodic flow?
2
votes
0answers
36 views

Ergodicity of differentiated processes

Let $S$ be a vector space, and $X$ a jointly-measurable random process/field with two parameters: $$ X: [0,\infty)\times\mathbb{R}\times\Omega\to S,$$ i.e. $X_{t,\theta}:\Omega\to S$ are random ...
5
votes
1answer
145 views

A counterexample for the Mean Ergodic Theorem in $L_\infty$

The so-called Mean Ergodic Theorem goes back to von Neumann for Hilbert spaces. Later on, versions of this result in reflexive Banach spaces have also appeared (see, e.g., the book by Krengel, Ergodic ...
11
votes
1answer
262 views

Is there a physical/geometric proof for L^2 boundedness of Bourgain's maximal function along the squares?

One problem that has bugged me for some time (though I only seriously thought about it for a month several years ago) is to give a physical proof of the L^2 boundedness of Bourgain maximal function ...
4
votes
1answer
228 views

Support of bivariate joint distribution of stationary and ergodic sequence

Let $\{X_t\}_{t\in \mathbb{N}}$ be a strictly stationary and ergodic sequence of real valued random variables and let the support of $X_1$ equal $[-1,1]$. Can the support of $(X_1,X_2)$ equal the unit ...
2
votes
2answers
132 views

special flows and Rudolph's theorem

The Rudolph's theorem confirm the existence of a special representation of an ergodic flow on the Lebesgue space. (In the book of I.P.Cornfeld entitled Ergodic theory). My question is: what is the ...
2
votes
2answers
151 views

iid random operator and its spectrum

consider an insteresting question: given Banach Space $ \mathcal{B}$, independent identical distribution random operator on $ \mathcal{B}$: $ (T_i)_{i \ge 1} $, where operator space is endowed with ...
2
votes
0answers
117 views

Average of irrational flow on the torus

Let $$F(x,y) = \frac{1}{\sqrt{2-\sin(2\pi x) - \sin(2\pi y)}}$$ defined on $\mathbb{T}^2$. Here $\mathbb{T}^2 = \mathbb{R}^2/ \mathbb{Z}^2$ is the 2-torus. How can I show that $$ \lim_{T\...
1
vote
0answers
46 views

Empirically random, quickly multiplicable matrices

I have encountered a need for fast computation of a transformation $Ax$ where $A\in \mathbb{C}^{K\times N},\ K\sim 10^7,\ N\sim 10^3$ is designed, and $x\in \mathbb{C}^N$ has iid $\mathcal{CN}(0,1)$ ...
3
votes
1answer
86 views

Measures maximizing entropy in a set of measures with fixed average for some observable

Let $\Omega$ be the set of all infinite binary sequences $(x_i)_{i\ge 0}$ endowed with the product topology coming from discrete topology on $\{0,1\}$. Consider $0<\alpha<1$ and let $$K_\alpha=\{...
4
votes
2answers
222 views

Lifting back the induced invariant measure / general version of Kac's formula for occupation times

Let $T$ be a conservative measure preserving (non-invertible!) transformation of a measure space $(X, \mathscr{F}, m)$ with infinite measure $m$. Let $A \in \mathscr{F}$ be such that $X = \cup_{k=0}^\...
-1
votes
1answer
102 views

Ergodicity of a measure preserving Anosov flow

Let $M$ be a Riemannian manifold and $\phi^t$ an Anosov flow on $M$. If $\phi^t$ is measure preserving (with respect to any Borel-measure on $M$), it is ergodic. Does anybody have a proof of that ...
3
votes
1answer
266 views

Unclear construction in a paper of Ornstein and Weiss

I originally posted this on math.stack, but no one answered, so im posting here: I need help understanding the following construction (Taken from the paper "Entropy and isomorphism theorems for ...
2
votes
0answers
78 views

Ergodicity in Césaro mean : deterministic and stochastic cases

Let $\{ X_t, t \geq 0 \}$ be a $\mathbb{R}^d$-valued stochastic process on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assumption $X_t$ is a regenerative process in the sense of https:/...
2
votes
0answers
30 views

Positive probability that two autoregressive sample paths have the same sign

Suppose I have two autoregressive processes of order one, that is, \begin{align} X_{t+1} &= \beta X_t + \varepsilon_{t+1}, \\ Y_{t+1} &= \beta Y_t + \varepsilon_{t+1}, \end{align} where $0<\...
0
votes
1answer
82 views

Positive upper asymptotic density and equidistribution

Let $B=\{b_n: n\geq 1\}$ be a set of positive integer numbers with positive upper asymptotic density and let $\alpha$ be a real irrational number. Is it true that $\{b_n \alpha\}$ is equidistributed ...
3
votes
1answer
74 views

Positive and Null recurrence of Markov Chains on a General State Space

Suppose $X_n$ is an irreducible, aperiodic and Harris recurrent Markov chain. It is well known that in this case, $X_n$ has a stationary distribution $\pi$. Are there any conditions that are ...
4
votes
2answers
74 views

Invariant function for Koopman operator of measure-class preserving tranformation

Let $(X,\mu)$ be a standard probability space and let $T:X \to X$ be a measure-class preserving transformation such that there is no $T$-invariant measure absolutely continuous with respect to $\mu$. (...
5
votes
1answer
182 views

Eigenvalue and eigenvector of ergodic Markov operator for continuous space Markov chain

As we know that the transition matrix $P$ of a Markov chain with finite space is a stochastic matrix, and from Perron-Frobenius Theorem, we know that the spectral radius of the matrix $P$ is $1$, and ...
1
vote
1answer
127 views

Understanding measure-preserving transformation [closed]

Given measure space $(S, \mathcal{S}, \mu)$, and measurable function $\phi: S \to S$. $\phi$ is measure-preserving if $\forall A \in \mathcal{S}, \mu(A) = \mu(\phi^{-1}(A))$. My confusion is that why ...
2
votes
1answer
165 views

Confusion about Teichmuller curves and $SL_2$ action

Let $M_g$ be the moduli space of curves, $\Omega M_g$ the total space of the bundle of holomorphic 1-forms and $\pi: \Omega M_g\to M_g$ the natural projection. On $\Omega M_g$ there's an action of $...
1
vote
0answers
45 views

Reference request- Automorphisms of point processes

A suspension of a point process on $\mathbb{R}^d$ is a measure preserving automorphism of the (distribution of the) point process which is determined by a map $T:\mathbb{R}^d\to\mathbb{R}^d$. The ...
2
votes
2answers
149 views

An analogue of the equidistribution theorem?

Suppose that $(n_k)_{k\in \mathbb{N}}$ is a given increasing sequence of positive integers. Does there exist an (irrational) number $a$ such that $\{an_k\}:=(a n_k)\text{mod }1 \rightarrow 1/2$ as $...
0
votes
1answer
99 views

How to compute the entropy of a random variable with values in a metric space? [closed]

I have a cloud of points, and I want to compute its 'diversity'. Variance is not appropriate, because a cloud clustering around few points can still have a large variance. To that end, I see the ...
12
votes
1answer
330 views

Entropy of composition

I asked this at math.stackexchange.com, but got no answers. Let $(X,B,\mu)$ be a probability space. Let $T,S:X→X$ be two measurable measure preserving maps that commute (i.e $TS=ST$). Let $A$ be a (...