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

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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 ...

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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, ...

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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 $\lambda:M\to \mathbb{R}$ ,$\lambda<1<m$.
...

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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 ...

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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 ...

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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 $\...

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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 ...

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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 $\...

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**1**answer

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}...

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votes

**1**answer

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 $p$...

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**1**answer

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 ...

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**1**answer

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 \...

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**1**answer

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 ...

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**2**answers

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 $\mu$
...

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**0**answers

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 }...

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**3**answers

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 ...

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**1**answer

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 ...

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**0**answers

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 ...

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**1**answer

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}, ...

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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 ...

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**0**answers

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 ...

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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 ...

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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 ...

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**0**answers

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 ...

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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,\...

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**2**answers

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?

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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 ...

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**1**answer

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 ...

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**1**answer

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 ...

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**1**answer

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 ...

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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 ...

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**2**answers

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 ...

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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\...

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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 $\mathcal{CN}(0,1)$ ...

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**1**answer

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=\{...

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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}^\...

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**1**answer

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 ...

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**1**answer

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 ...

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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:/...

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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<\...

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**1**answer

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 ...

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**1**answer

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 ...

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**2**answers

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 $\mu$. (...

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**1**answer

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 ...

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**1**answer

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 ...

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**1**answer

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 $...

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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 ...

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**2**answers

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 $...

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votes

**1**answer

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 ...

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**1**answer

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 (...