Fully chaotic conservative models for some torus homeomorphisms

Abstract

We study homotopic-to-the-identity torus homeomorphisms, whose rotation set has nonempty interior. We prove that any such map is monotonically semiconjugate to a homeomorphism that preserves the Lebesgue measure, and that has the same rotation set. Furthermore, the dynamics of the quotient map has several interesting chaotic traits: for instance, it is topologically mixing, has a dense set of periodic points and is continuum-wise expansive. In particular, this shows that a convex compact set of ${\mathbb{R}}^2$ with nonempty interior is the rotation set of the lift of a homeomorphism of ${\mathbb{T}}^2$ if and only if it is the rotation set of the lift of a conservative homeomorphism.

Minor expository modifications following referee suggestions. To appear in Memoirs of the American Mathematical Society. 100 pages, 22 figures