Some Properties of ASH attractors

Abstract

This text refers to a doctoral thesis in dynamical systems, which focuses on continuous-time dynamics. More specifically, it advances the theory of asymptotically sectional-hyperbolic (ASH) flows through several fundamental results. First, we prove that every star ASH attractor of a C¹ vector field with positive topological entropy is necessarily sectional-hyperbolic. Second, we establish that all ASH attractors satisfy the intermediate entropy property. Third, we demonstrate that any ASH attractor in three-dimensional vector fields is entropy-expansive and admits periodic orbits. Finally, we provide a lower bound for the growth rate of periodic orbits in an ASH attractor.