Lifting generic points

TL;DR

The paper generalizes a theorem on lifting generic points in topological dynamical systems with the weak specification property, extending results from finite shifts to more general systems.

Contribution

It extends Kamae's theorem from finite shifts to systems with the weak specification property, showing the existence of generic points in product systems.

Findings

Existence of generic points in product systems with weak specification.

Generalization of Kamae's theorem to broader classes of dynamical systems.

Applicable to systems beyond finite alphabet shifts.

Abstract

Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $\xi$ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $\mu$ on $X$ and $\nu$ on $Y$, with $\mu$ ergodic. Let $y\in Y$ be quasi-generic for $\nu$. Then there exists a point $x\in X$ generic for $\mu$ such that the pair $(x,y)$ is quasi-generic for $\xi$. This is a generalization of a similar theorem by T.\ Kamae, in which $(X,T)$ and $(Y,S)$ are full shifts on finite alphabets.