Filtrations, hyperbolicity and dimension for polynomial automorphisms of C^n

Abstract

In this paper we study the dynamics of regular polynomial automorphisms of C^n. These maps provide a natural generalization of complex Henon maps in C^2 to higher dimensions. For a given regular polynomial automorphism f we construct a filtration in C^n which has particular escape properties for the orbits of f. In the case when f is hyperbolic we obtain a complete description of its orbits. In the second part of the paper we study the Hausdorff and box dimension of the Julia sets of f. We show that the Julia set J has positive box dimension, and (provided f is not volume preserving) that the filled-in Julia set K has box dimension strictly less than 2n. Moreover, if f is hyperbolic, then the Hausdorff dimension of the forward/backward Julia set J^{+/-} is strictly less than 2n.