07/09/2016, 10:20 — 11:00 — Amphitheatre Pa2, Mathematics Building
Alberto Pinto, Universidade do Porto

Explosion of smoothness for conjugacies

Let $f$ and $g$ be piecewise $C^r$ maps of the interval, with $r > 1$ and non-flat at the discontinuity sets $C_f$ and $C_g$, respectively, and let $h$ be a topological conjugacy between $f$ and $g$. We note that the maps $f$ and $g$ can be discontinuous and/or have different lateral derivatives (zero, finite or infinite) at the non-flat discontinuity sets $C_f$ and $C_g$, respectively. Let $A$ be a cycle of intervals of $f$, whose $supp A$ is a chaotic topological attractor. We prove that, if $h$ is $C^1$ in a single point with non-zero derivative then the conjugacy is a $C^r$ diffeomorphism.