Resumen de Estimates for the stable dimension for holomorphic maps
Mariusz Urbanski, Eugen Mihailescu
We study the Hausdorff dimension of the intersection between stable manifolds and basic sets for an Axiom A holomorphic endomorphism on the complex projective space of dimension 2. We improve an upper estimate given in a previous paper by Mihailescu, by taking into consideration the number of preimages, and thus proving for non-invertible maps results parallel to those of Verjovsky and Wu from the case of Henon diffeomorphisms. Also, a lower estimate for the above stable dimension is given by using a concept of preimage entropy modeled after Bowen. If the map is not a homeomorphism, then the preimage entropy may not coincide with the usual forward entropy. We also construct examples of holomorphic Axiom A maps which are injective on their respective basic sets and such that their stable dimension is strictly positive. We study in the end the stable dimension for a class of special quadratic endomorphisms.
