The space of invariant measures of Kan-like diffeomorphisms
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| Bárbara Nuñez Madariaga |
| Pontificia Universidad Católica de Valparaíso |
| Abstract |
| In 1994, Ittai Kan proposed a strategy to prove that the partially hyperbolic diffeomorphism \(K:\mathbb{R}^2\times [0,1]\to \mathbb{T}^2\times [0,1]\) defined by \[K(x,y,t)=\left(3x+y, 2x+y,t+\frac{1}{32}t(1-t)\cos(2\pi x)\right)\] |
| has exactly two physical measures with intermingled basins. In other words, 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 presentation, we will analyze the fundamental components that generate this phe- nomenon and demonstrate how they also apply to similar behaviors of the basins corresponding to different invariant measures in Kan-like applications, such as the measure of maximal entropy. I work in progress with Lorenzo Diaz (PUC-Rio), Katrin Gelfert (UFRJ), and Carlos Vasquez (PUCV) (see, [1, 2]). |
| References |
| [1] Vaughn Climenhaga, Yakov Pesin, and Agnieszka Zelerowicz. Equilibrium states in dynamical systems via geometric measure theory. Bull. Amer. Math. Soc. (N.S.), 56(4):569–610, 2019. |
| [2] Bárbara Núñez Madariaga, Sebastián A. Ramírez, and Carlos H. Vásquez.829 Measures maxi- mizing the entropy for Kan endomorphisms. Nonlinearity,830 34(10):7255–7302, 2021. |
