The set K^- for hyperbolic non-invertible maps

Abstract

Any Axiom A holomorphic map on \mathbb{P}^2\mathbb{C} has a natural ergodic measure \mu. The set K^- is the set of points which have arbitrarily close neighbors with prehistories not convergent to supp \mu.

This set is the analogue of the set of points with bounded backwards iterates from the case of Hénon mappings. For Hénon diffeomorphisms it was shown by Bedford and Smillie that K^- either has an empty interior or \mathop{K}\limits^\circ{}^{-}= union of the basins of finitely many repelling periodic points. For s-hyperbolic holomorphic endomorphisms of \mathbb{P}^2, we show here that the only possibility is \mathop{K}\limits^\circ{}^{-} = \emptyset. This answers a question of Fornaess.

We also prove that \mathop{K}\limits^\circ{}^{-} is the union of finitely many repelling basins when the endomorphism is a perturbation of a Hénon mapping.

Several non-trivial examples of s-hyperbolic maps are discussed as well, some of them coming from perturbations of product maps and others from solenoids.