Polynomial diffeomorphisms ofC 2. IV: The measure of maximal entropy and laminar currents

Abstract

The simplest holomorphic dynamical systems which display interesting behavior are the polynomial maps of C. The dynamical study of these maps began with Fatou and Julia in the 1920's and is currently a very active area of research. If we are interested in studying invertible holomorphic dynamical systems, then the simplest examples with interesting behavior are probably the polynomial diffeomorphisms of C 2 . These are maps f : C 2 → C 2 such that the coordinate functions of f and f −1 are holomorphic polynomials.For polynomial maps of C the algebraic degree of the polynomial is a useful dynamical invariant. In particular the only dynamically interesting maps are those with degree d greater than one. For polynomial diffeomorphisms we can define the algebraic degree to be the maximum of the degrees of the coordinate functions. This is not, however, a conjugacy invariant. Friedland and Milnor [FM] gave an alternative definition of a positive integer deg f which is more natural from a dynamical point of view. If deg f > 1, then deg f coincides with the minimal algebraic degree of a diffeomorphism in the conjugacy class of f . As in the case of polynomial maps of C, the polynomial diffeomorphisms f with deg(f ) = 1 are rather uninteresting. We will make the standing assumption that deg(f ) > 1.For a polynomial map of C the point at infinity is an attractor. Thus the "recurrent" dynamics can take place only on the set K consisting of bounded orbits. A normal families argument shows that there is no expansion on the interior of K so "chaotic" dynamics can occur only on J = ∂K. This set is called the Julia set and plays a major role in the study of polynomial maps.For diffeomorphisms of C 2 each of the objects K and J has three analogs. Corresponding to the set K in one dimension, we have the sets K + (resp. K − ) consisting of the points whose orbits are bounded in forward (resp. backward) time and the set K := K + ∩ K − consisting of points with bounded total orbits. Each of these sets is invariant and K is compact. As is in the one dimensional case, recurrence can occur only on the set K.Corresponding to the set J in dimension one, we have the sets J ± := ∂K ± , and the set J := J + ∩ J − . Each of these sets is invariant and J is compact. A normal families argument shows that there is no "forward" instability in the interior of K + and no "backward" instability in the interior of K − . Thus "chaotic" dynamics, that is recurrent dynamics with instability in both forward and backward time, can occur only on the set J.The techniques that Fatou and Julia used in one dimension are based on Montel's theory of normal families and do not readily generalize to higher dimensions. A different tool appears in the work of Brolin [Br], who made use of the theory of the logarithmic 1

On loops in the hyperbolic locus of the complex Hénon map and their monodromies

“…By the result of Bedford, Lyubich and Smillie [3,Theorem 10.1], we have H R ⊂ H C ∩ R 2 . It is then natural to ask what happens in the rest of H C ∩ R 2 .…”

Some remarks on the connectivity of Julia sets for 2-dimensional diffeomorphisms

“…By Poincaré recurrence, we may suppose that for infinitely many n j , f n j (x) ∈ P , so by using the stable manifold theorem, we get that for large j, f n+nj (V ) contains a disk arbitrarily close to W u loc (f nj x). Now if p is any saddle point, W s (p) has transverse intersection points with any set of positive measure of unstable disks [BLS,Lemma 9.1]. We conclude that W s (p) must intersect f n+nj (V ), hence V , transversely.…”

“…Notice that the current T + induces by restriction a measure on every unstable leaf. These measures are equivalent to the "unstable conditionals" of µ [BLS,Prop 3.1].…”

“…and we get that for large n, f n (V ) has intersection points with a set of positive measure of disks in S + and most of them are transverse [BLS,Lemma 6.4]. Assume f n (V ) intersects W s loc (x), for some x ∈ P , not necessarily periodic.…”

“…Consider a Pesin box P of positive measure; the local stable manifolds associated to points form a piece of stable lamination of positive transverse measure, hence a uniformly laminar current [BLS,§8] …”