Articles of ergodic theory

Weak convergence of finite measure preserving transformations

I am reading King’s paper “The commutant is the weak closure of the powers, for rank-1 transformation” and I am not able to show that: (0.1) “If the $T_i$ are invertible measure preserving transformations on Lebesgue probability space, commuting with each other and converging weakly to $S$ then $S$ is invertible and $T_i^{-1}\rightarrow S^{-1}$ weakly.” […]

Why every strict stationary process have the following representation

Suppose $\{X_n\}_{n\geq 1}$ is a strict stationary process, that is for every $n$, $(X_0,\ldots,X_n)$ and $(X_1,\ldots,X_{n+1})$ have the same distribution. Then there is a probability space $(\Omega,\mathcal{F},P)$, a r.v. $X$ and a measure-preserving transformation $T$ on $\Omega$ such that $X_n(\omega)= X(T^n(\omega))$ holds for all $n$. I wonder why this is true when the $\{X_n\}$ have […]

A generic point for a non-ergodic measure

Definition: Let $X$ be a compact metric space and let $\mu$ be a Borel probability measure on $X$, then we say that the sequence $(x_n)$ of elements of $X$ is equidistributed with respect to $\mu$ if we have that for any $f \in C(X)$ $$\frac1n \sum_{m = 1}^n f(x_m) \to \int_X f(x) \, d\mu(x).$$ Now, […]

convergence of entropy and sigma-fields

This question is related to this one. Let $(X_1, X_2, \ldots)$ be a sequence of random variables such that each $X_n$ takes its values in a finite space, say $\{0,1\}$, and the $\sigma$-field $\sigma(X_1, X_2, \ldots)$ is discrete (generated by a denumerable partition of events). The maximal possible entropy of $(X_1, \ldots, X_n)$ is $n […]

Topologizing Borel space so that certain functions become continuous

Let $X$ and $Y$ be compact metric spaces. Let $f:X \to Y$ be a Borel measurable map and suppose that $T:X \to X$ is a homeomorphism. Can one change the topology on $X$ such that $X$ is still a compact metrizable space, with the same Borel sets. The map $T$ remains continuous. The map $f:X […]

Recurrent point of continuous transformation in a compact metric space

Given a metric space $(X,d)$ and a transformation $T:X\rightarrow X$, a point $x\in X$ is said to be recurrent iff it belongs to the closure of its orbit $\{T(x), T^2(x),…\}$: more precisely, there exists an increasing sequence $(n_k)$ of natural numbers with $n_k \rightarrow \infty$ such that $T^{n_k}(x)\rightarrow x$ when $k \rightarrow \infty$. Show that […]

Lemma 1.7 in chapter 5 in Ergodic Theorems by Ulrich Krengel.

The lemma is on page 181. Here’s the lemma and its proof: I don’t understand how did they get: $$\lim | S^{n_k+1}f(\omega)|\leq \gamma (\alpha – \epsilon)+(1-\gamma)\alpha$$ The term that is confusing is $\gamma (\alpha – \epsilon)$, I mean I understand why $\alpha – \epsilon$, but not why times $\gamma$. I mean $P(\omega_k , F ) […]

Fractional part of $n\alpha$ is equidistributed

Let $\alpha$ be an irrational number. Then the sequence $\{\{n\alpha\}\}$ is equidistributed. I am using the following definition of equidistribution. A sequence $\{a_i\}$ is equidistributed if $\frac1n\sum\limits_{i=1}^n f(a_i) \to \lambda(f):=\int_0^1 f(x)\,dx$ for all continuous $f:[0,1]\to \mathbb{R}_{\ge 0}$. In the proof I used the fact that if $T_g(x)=(g+x) \pmod 1$, then $\lambda(f\circ T_g)=\lambda(f)$ for all continuous […]

Simultaneous Diophantine approximation: multiple solutions required

Suppose we have $n$ linearly independent (over $\mathbb{Q}$) irrational numbers $\{ \alpha_i | 1\leq i \leq n \}$. For the simultaneous Diophantine approximation problem $$ |q \alpha_i – p_i | < \epsilon , $$ where $q$ and the $p$’s are all integers, we have the LLL algorithm. The problem is, by this algorithm, for each […]

Question about entropy

Let $(X,A,\nu)$ be a probability space and $T\colon X\to X$ a measure preserving transformation $\nu$. Take a measurable partition $P=\{P_0,\dots,P_{k-1}\}$. Let$I$ be a set of all possible itineraries, that is, $I=\{(i_1,\dots,i_n,\dots)\in k^N;$ there is a $x\in X$, such that $T^n(x)\in P_{i_n}$ for all $n\in\Bbb N$. Suppose that $I$ is countably infinite. Is true that the […]