Estimates for the stable dimension for holomorphic maps

• Autores: Mariusz Urbanski, Eugen Mihailescu
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 31, Nº 2, 2005, págs. 367-390
• Idioma: inglés
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• Resumen

  • 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.