Mixing completely scrambled system exists

TL;DR

This paper proves the existence of topologically mixing, completely scrambled homeomorphisms on various spaces, solving a long-standing open problem in dynamical systems.

Contribution

It establishes the existence of mixing, completely scrambled homeomorphisms on spaces of any dimension, addressing a 15-year-old open question.

Findings

Existence of topologically mixing, completely scrambled homeomorphisms.

Construction of such homeomorphisms on spaces of arbitrary dimension.

Resolution of a 15-year-old open problem in topological dynamics.

Abstract

We prove that there exists a topologically mixing homeomorphism which is completely scrambled. We also prove that for any integer $n\geq 1$ 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.