Question about probability spaces $(X,\mathcal B,\mu)$ with a measurable map $T\colon X\to X$ preserving the measure, that is $\mu(T^{—1}A)=\mu(A)$ for all $A$ measurable.

learn more… | top users | synonyms

1
vote
1answer
61 views

Ergodicity vs Weak Mixing

I have been trying to prove the following: Let $(\Omega,\mathcal{F})$ be a measurable space endowed with probability measure $\mathbb{P}$. Suppose $\tau : \Omega \to \Omega$ is a measure preserving ...
1
vote
1answer
28 views

Birkhoff Ergodic theorem for two measures

Suppose $(X,\mathcal{B}, \mu, T)$ and $(X,\mathcal{B}, \nu, T)$ are both ergodic ppt. I'm a bit confused how the B Ergodic Theroem works since the LHS of the equation doesn't involve $\mu$ or $\nu$, ...
1
vote
1answer
29 views

Wandering set definition

I've seen two apparently different definitions and am wondering which is correct. A set $W$ is wandering if $\{T^{-k}W; k\in \mathbb{N}_0\}$ (resp. $\{T^{k}W; k\in \mathbb{N}_0\}$) are pairwise ...
0
votes
0answers
185 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
0
votes
1answer
27 views

Using Furstenberg's skew-product for $\alpha n^2$ equidistributed.

I am having trouble proving that if $\alpha_1,\alpha_2$ are irrational, then the sequence $(\alpha_1n,\alpha_2n^2)$ is equidistributed in $\mathbb{T}^2$. It is straightforward to use Furstenberg's ...
2
votes
1answer
27 views

Approximation of integrals smooth functions via sampling with non-Liouville number.

Suppose $\alpha$ irrational is not a Liouville number. i.e. we cannot find an a sequence of rational approximations $p_n/q_n$ with $|\frac{p_n}{q_n}-\alpha|<\frac{1}{q_n^n}$. I am trying an ...
2
votes
2answers
27 views

Consequences of Arnoux, Ornstein, Weiss Theorem.

The theorem states that any invertible, aperiodic, measure-preserving system on a Borel probability space is isomorphic to a cutting and stacking transformation. My question is, why is this useful? ...
2
votes
1answer
30 views

Rotation of the torus $T^2$ by irrational numbers linearly dependent over $\mathbb Z$

It is known that the rotation $x \to x + \alpha$ of $S^1 = \mathbb R / \mathbb Z$ with irrational $\alpha$ is ergodic and, in particular, $\alpha n$, $n = 1, 2,\dots$, are dense in $S^1$. In two ...
0
votes
2answers
102 views

Initial conditions that converge to an unstable equilibrium

Consider the discrete-time dynamical system \begin{align} x_{k+1}=T(x_k), \end{align} where $k\in\mathbb{N}$, $x_k\in\mathbb{R}^n$, $x_0$ is the initial condition, and $T:\mathbb{R}^n\to\mathbb{R}^n$ ...
2
votes
0answers
32 views

Spectrum of $T^2$

Is it possible that $T^2$ has a discrete spectrum when $T$ is an invertible measure-preserving transformation whose spectrum is continuous or mixed ?
1
vote
1answer
21 views

can two Markov kernels be close in total variation and differ in their ergodicity properties?

This is a question inspired by a recent MCMC paper and linked to an earlier question of Sam Livingstone that did not get any answer. Given two Markov kernels $\mathfrak{K}$ and $\mathfrak{H}$ such ...
2
votes
1answer
37 views

Ergodicity of the natural measure implies uniqueness of the invariant density?

Consider a dynamical system $x_{t+1} = F(x_t)$ defined in $\Omega$ and its natural measure to be $\mu$. The Perron-Frobenius operator $F$ maps the density $f$ in time according to $f_{t+1} = F(f_t)$. ...
1
vote
0answers
25 views

Why does ergodicity not neccesarily imply ergodic for the mean?

I'm trying to answer a question where I have an ergodic and covariance stationary process $\{x_t\}$, and without imposing further moment conditions need to prove $\frac{1}{n} \sum\limits_{t=1}^n x_t^2 ...
1
vote
1answer
52 views

Ultrafilter and upper natural densities

It is straightforward to show that there is an ultrafilter $\mathcal{U}_0$ on the positive integers such that every element $A\in \mathcal{U}_0$ satisfies $$d^\ast(A):=\limsup_{n\to +\infty} \frac{\...
5
votes
2answers
101 views

An intuitive explanation of how the mathematical definition of ergodicity implies the layman's interpretation 'all microstates are equally likely'.

I'm self-studying Statistical Mechanics; in it I got Fundamental Postulate of Statistical Mechanics and that took me to ergodic hypothesis. In the most layman's language, it says: In an isolated ...
2
votes
0answers
32 views

Moving Average of an Ergodic Markov processes.

Let $ \{X(t); t\geq 0\} $ be an ergodic Markov process, and let G be a positive integrable function with $\int_0^\infty G(x)dx=1$. Does $$F(t)\doteq G*X(t) = \int_0^t G(t-s)X(s)ds $$ converge in ...
3
votes
0answers
110 views

Zeta functions for flows on hyperbolic surfaces

The Riemann zeta function is defined by $$ \zeta(s)=\sum_{n=1}^\infty \frac1{n^s} $$ and can be written in the form $$ \zeta(s)=\prod_p\frac1{1-p^{-s}}, $$ where the product is over all prime numbers $...
4
votes
0answers
26 views

Question about a type of continuous state Markov-process.

EDIT: Solved! It turns out that if the function is continuous and various regularity conditions hold then the statement is true. This has been established in the 'stochastic approximation' literature, ...
0
votes
1answer
33 views

multiples of subset of $\mathbb{N}$

Suppose that $ A_1\cup \dotsm A_n=\mathbb{N}$ is a partition of $\mathbb{N}$ into disjoint subsets. Is it true that there is an integer $1 \leq k \leq n$ such that the set $A_k\cap 2A_k$ is infinite?
1
vote
0answers
37 views

Bernoulli-Shift is a stationary process

Let be $(\Omega, \mathcal{F}, \mathbb{P})=([0,1), \mathcal{B}([0,1)), \lambda)$. We define $Y_n : \Omega \rightarrow \Omega$ by $Y_n := 2Y_{n-1} \mod 1$. Many sources claim that this is a stationary ...
1
vote
0answers
30 views

Invariant $\sigma-$ field of double infinite stationary process

We know any stationary process ${(X_n)}_{n \in \mathbb{Z}}$ can be represented as $X_n(\omega)=X(\phi^n\omega)$. where we take the shift $\phi$ on the canonical space of the process and $X$ maps a ...
0
votes
0answers
29 views

Non-integrability and splitting of separatrices

It is well-known that the (first-order) Melnikov method is the standard technique to detect non-integrability of a perturbed system of ordinary differential equations or maps. Namely, the unperturbed ...
2
votes
0answers
22 views

Deterministic time changed ergodic process

This is more of a "ask-for-idea" than "ask-for-answer" question: Suppose $\{X_t\}$ is an ergodic process with a known stationary\limiting distribution $\pi$. Let $f(t)$ be a deterministic and ...
2
votes
0answers
60 views

Convergence of empirical average of Markov chain from transient class

I am trying to get an intuition of how to understand the limit of the empirical average $$\frac1n\sum_{i=1}^nX_i\tag{$\ast$}$$ of some Markov chain $(X_n)_n$ with transition matrix $P$ (let's assume ...
3
votes
0answers
18 views

Is there a mixing condition to get the decay property I want?

Let $(X,\mu)$ be a probability measure space and $T:X\to X$ an ergodic invertible measure preserving transformation. Consider a measurable set $A\subset X$ with $0<\mu(A)<1$ For each $N$ define ...
4
votes
2answers
50 views

Minimal distance for Irrational Rotations on the circle

It is a well known fact that for $\alpha$ irrational that $\langle n\alpha\rangle$ is dense on the unit circle. I want to know what the result is for computing $a_n=\min_{1\le N \le n} |\langle N \...
6
votes
1answer
64 views

Sawyer's proof that a certain lim sup of sets has full measure

Given $(X, \mathcal{A}, \mu)$ a probability space, let $\mathcal{F}$ be a family of $\mu$-invariant measurable functions, closed under composition, with the following property: If $A$ is a measurable ...
1
vote
0answers
89 views

Testing whether a particular set of measures borelianas is a set of Baire

Let $X$ compact metric space and $F:X\times \mathbb{R}\rightarrow X$ flow continuous ($F(x,t)=F_t(x)$). If $\delta>0$ we define $$\Lambda(x,\delta)=\bigcup_{h\in B}\bigcap_{t\in\mathbb{R}}F_{-h(t)...
0
votes
0answers
30 views

expected value of a doubly stochastic matrix with i.i.d entries

I am now thinking a problem: what is the expected value of doubly stochastic matrix with i.i.d entries Each entries is i.i.d in $[0,1]$. Will the answer be a matrix with all entries $\...
2
votes
1answer
74 views

Cesaro summation

Let's consider $\{a_{n} \} -$ a bounded sequence of real numbers. Is it true that $\frac{1}{n} \sum_{k=0}^{n-1}{|a_{k}|^{p}}$ (Cesaro sums) converges or diverges for all $p \geq 1$? (more presicely: ...
4
votes
0answers
77 views

Equidistribution theorem of Weyl

Have you examples of applications of Equidistribution theorem of Weyl in proofs of irrationality of numbers? I don't know if "if and only if" is true for this theorem.
0
votes
0answers
9 views

Varying measures and invariant measures

Are there any results related to varying measures? Why are invariant measures so useful/important?
0
votes
0answers
34 views

Ergodic means and Birkhoff theorem

Let's consider the following map $$F(x, y) = \lim_{n \to \infty}{\frac{1}{n} \sum_{k=0}^{n-1}{f(\{x + ky \})}}$$ and $f(x) = x(1-x)$. I would like to evaluate the value of $F(x, y)$ for arbitrary ...
1
vote
0answers
21 views

Confusion about the concept of ergodicity in random process

This is a basic question I came across when I started learning random processes. Suppose I have a random variable $f(t) = G(t) + n(t)$ where $G(t) = A\exp(-t^2/T^2)$ (i.e. a deterministic function ...
4
votes
0answers
36 views

Reference on the history of ergodic theory

I'm looking for some good books on the history of ergodic theory. I'm a Ph.D student in the field, and I am taking Steven Weinberg's advice to learn about the history of my field: http://math....
1
vote
0answers
29 views

Strong mixing property for endomorphisms of a finite set

Let's consider $X = \{1, 2, \ldots, n \}$. I would like to establish, how many of the maps $f: X \to X$ have the following strong mixing property: For a given triple $(X, \mathcal{B}, \mu)$, $T: X ...
0
votes
1answer
82 views

Eventually periodic continued fraction implies root of polynomial of degree 2

How to prove that every irrational number with eventually periodic continued fraction expansion is a root of a polynomial of degree 2?
1
vote
2answers
69 views

Can a mixing process be non-stationary?

I was always under the impression that a mixing process is ergodic and an ergodic process is necessarily stationary, so that a mixing process is stationary. I have come across a paper discussing non-...
1
vote
0answers
51 views

An aquidistributed sequence

Prove that $\{an^\sigma\}$ is equidistributed in $ [0,1) $,if $\sigma>0$ is noninteger and $a\neq 0$. I know how to solve this problem if $\sigma <1$ , so it is not a duplicate of ...
2
votes
0answers
32 views

Ergodicity under measure-theoretic isomorphism

Suppose we have two measurable dynamical systems $(X_1,\mathcal{B}_1,\mu_1,T_1)$ and $(X_2,\mathcal{B}_1,\mu_2,T_2)$, with $\mu_i(X_i)=1,\ i=1,2$. Suppose they are measure-theoretically isomorphic (...
2
votes
2answers
45 views

why topological conjugacy does not preserve ergodicity?

I want to know why topological conjugacy does not transfer ergodicity from one dynamical system to another?and if we change the maps from continous to C1-differentiable will the problem solve or not?
2
votes
1answer
41 views

$T(x) = \{\frac{1}{x} \}$ invariant measure

Let's consider $T: (0,\hspace{-0.05 in}1] \rightarrow [0,\hspace{-0.05 in}1)$ which is defined as $Tx = \{ \frac{1}{x} \}$. It can be shown that Lebesgue measure is not invariant under this map. ...
2
votes
0answers
35 views

Ergodic theorem in two variables?

I just started learning ergodic theorem due to the need in a research project. I am aware of the following form of ergodic theorem: If $\{X_n\}$ is an ergodic process with state space $\mathcal{X}$ ...
1
vote
0answers
39 views

Uniform integrability of functional of an ergodic Markov Chain?

Suppose $\{X_n\}$ is an ergodic Markov Chain with general state space $\mathcal{X}$ and stationary distribution $\pi$. Suppose $\forall x\in\mathcal{X}, f(x)\geq 0$ and $\mathbb{E}_{\pi}[f(X)]<\...
6
votes
2answers
69 views

A discontinuous almost everywhere map does not admit an invariant measure

Let's consider a map $T: X \rightarrow X$ so that it's discontinuous almost everywhere (in particular, let $X = \mathbb{R}$, and $T = 1_{\mathbb{Q}}$ -- Dirichlet function). Is it true that $T$ does ...
0
votes
1answer
21 views

Integrability and area-preservation property of maps

Suppose we have a map defined on a plane, $x_{1}=f(x_{0})$, where $x \in \mathbb{R}^{2}$. Assume it is integable: there exists a function $I$ of the phase space variable $x$ such that $I(x)=I(f(x))$. ...
0
votes
1answer
45 views

Rotation map on $S^1$ preserves measure

I'm having a little trouble understanding following the example in my book as to why the rotation map $R_{\alpha}$ preserves Lebesgue measure. We have $R_{\alpha}([x])=[x+\alpha]$ and $R_{\alpha}^{-...
1
vote
0answers
40 views

Question about dense trajectory on $k$-dimensional torus under rotation map

Today when I was doing ergodic theory problems I faced with following problem: Assume rotation map on $k$-dimensional torus under $\alpha=(\alpha_1,...,\alpha_n)$ then orbit of all $x$ in $k$-...
1
vote
1answer
39 views

Computing Lyapunov Exponents

Consider $\mathcal{T}=S^{1}\times D^{2}$, where $S^{1}=[0,1]\mod 1$ and $D^{2}=\{(x,y)\in\mathbb{R}^{2}|x^{2}+y^{2}\le 1\}$. Fix $\lambda\in(0,\frac{1}{2})$ and define the map $F:\mathcal{T}\to\...
1
vote
1answer
46 views

Question About Definition of Lyapunov Exponents

I just have a quick question about the definition of Lyapunov exponents. My textbook defines them for a smooth map $f:M\to M$, where $M$ is a smooth manifold. For $x\in M$ and $v\in T_xM$: $\lambda(x,...