# Tagged Questions

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.

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### 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 ...
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### 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 $\nu$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 $T:\mathbb{R}^n\to\mathbb{R}^n$ ...
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### 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 ?
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### 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 ...
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### 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 $f_{t+1} = F(f_t)$. ...
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### 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: ...
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### 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.
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### Varying measures and invariant measures

Are there any results related to varying measures? Why are invariant measures so useful/important?
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### 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 ...
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### 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 ...
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### 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....
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### 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 ...
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### 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 $I(x)=I(f(x))$. ...
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 $R_{\alpha}^{-... 0answers 40 views ### Question about dense trajectory on$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$-... 1answer 39 views ### Computing Lyapunov Exponents Consider$\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\...
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,...