Furstenberg type theorems for strongly transitive maps

Abstract

Given a dynamical system $(X,f)$ and a positive integer n, we study the concept of strong transitivity in the induced map $f^{\times n}:X^n\to X^n$. The main problem is to figure out when the strong transitivity of $f^{\times 2}$ implies the strong transitivity of $f^{\times n}$. In certain spaces, such as finite connected graphs, we show that if $f^{\times 2}$ is strongly transitive then f is strongly mixing which also implies that $f^{\times n}$ is strongly transitive for any n. We also give an example of a map $f:S^1\to S^1$ which is strongly mixing but not exact, showing that our result is optimal in finite graphs.