TOPOLOGICAL ENTROPY ON SADDLE SETS IN P 2 JEFFREY DILLER and MATTIAS JONSSON

0. Introduction. The Fatou set of a holomorphic map f : P is the largest open subset of P on which iterates of f form a normal family. The complement of is called the Julia set when k = 1, and it is well known that the Julia set is the closure of the set of repelling periodic points. When k = 2, however, even product maps suffice to show that the structure of P2\ is more intricate. For instance, P2\ contains both repelling periodic points and periodic points of “saddle” type, with one expanding and one contracting direction. In nice situations, these distinct types of periodic points occupy distinct regions in P2 \ , and each of these smaller regions legitimately vies with P2 \ for the designation of Julia set. Our concern in this paper is with what we call saddle sets of a holomorphic map f : P2 . These generalize the notion of a saddle periodic point, and while we defer the precise definition until Section 1, the following description should suffice for the moment. A closed invariant set = f ( ) ⊂ P2 is a saddle set of f if f acts transitively and hyperbolically on with one contracting and one expanding direction, and if is in some sense both maximal and isolated as a hyperbolic set. Important examples of saddle sets are given by the basic sets of saddle type for an Axiom A map f : P2 . Indeed, the paper [FS2] of Fornæss and Sibony on Axiom A holomorphic maps of P2 inspired much of the work on which this paper is based. Given a history p̂ = (pj )j≤0 in (i.e., pj ∈ and f (pj−1) = pj for all j ≤ 0) and a small fixed δ > 0, the associated local unstable manifold is