Sinks with relatively large immediate basins and a refinement of Mañé’s C1 generic dichotomy

Abstract

A C1 perturbation theorem creating sinks with relatively large immediate basins of attraction under the existence of a non-atomic ergodic measure admitting at most small positive Lyapunov exponents is provided. As an application, we prove a refinement of Mañé’s C1 generic dichotomy for surface diffeomorphisms, that is, C1 generically they have either (I) hyperbolicity; or (II) (by taking the inverse if necessary) infinitely many attracting periodic orbits each of which has either (a) relatively large immediate basins of attraction or (b) a pathological feature (i.e. the maximal and minimum norms of the derivatives along the periodic orbits increase and decrease exponentially, respectively, at the periods by a uniform rate). Moreover, the property (a) of (II) is considered from numerical viewpoints in the context of ‘observability’.