Filtrations, hyperbolicity and dimension for polynomial automorphisms

In this paper we study the dynamics of regular polynomial automorphisms of C. These maps provide a natural generalization of complex Hénon maps in C to higher dimensions. For a given regular polynomial automorphism f we construct a filtration in C 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.