On the Structure of the Space of Invariant Measures for Kan-Type Diffeomorphisms

Resumen: In 1991, Ittai Kan proposed a strategy to prove that the diffeomorphism $K\colon \mathbb T^2\times[0,1]\to \mathbb T^2\times[0,1]$ defined by

$$K(x,y,t) = (3x + y, 2x + y, t + \tfrac{1}{32}t(1 - t)\cos(2\pi x))$$

has exactly two physical measures whose basins are intermingled; that is, every open set in the ambient space contains a Lebesgue-positive set of points belonging to the basin of each of the physical measures.

In this talk, we will unravel the key elements that give rise to this phenomenon and show how these same elements also account for similar behaviors in the basins of other invariant measures for $K$, such as the measure of maximal entropy. We will present some partial results obtained in collaboration with Bárbara Nuñez-Madariaga (PUCV), Katrin Gelfert (UFRJ), and Lorenzo Díaz (PUC-Rio).