Expansivness is a fundamental property of dynamical systems.
It is sometimes viewed as an indication to chaos.
However, expansiveness also sets limitations on the complexity of a system.
Ma\~{n}'{e} proved in the 1970’s that a compact metric space that
admits an expansive homeomorphism is finite dimensional.
In this talk we will discuss a recent extension of Ma\~{n}'{e}’s
theorem for actions generated by multiple homeomorphisms,
based on joint work with Masaki Tsukamoto. This extension relies on a
notion called “topological mean dimension’’ , introduced by Gromov and
developed by Lindenstrauss and Weiss in the late 1990’s.