Mixing completely scrambled system exists

Abstract

We prove that there exists a topologically mixing homeomorphism which is completely scrambled. We also prove that for any integer $n\ge 1$ there is a continuum of topological dimension $n$ supporting a transitive completely scrambled homeomorphism, and $n$-dimensional compactum supporting a weakly mixing completely scrambled homeomorphism. This solves a 15 year old open problem.

To appear in Ergodic Theory and Dynamical Systems