Polynomial Diffeomorphisms of C 2 . Viii: Quasi-expansion

نویسنده

  • John Smillie
چکیده

This paper continues our investigation of the dynamics of polynomial diffeomorphisms of C 2 carried out in [BS1-7]. There are several reasons why the polynomial diffeomorphisms of C 2 form an interesting family of dynamical systems. Not the least of these is the fact that there are connections with two other areas of dynamics: polynomial maps of C and diffeo-morphisms of R 2 , which have each received a great deal of attention. The fact that these three areas are linked makes it interesting to understand different dynamical notions in these three contexts. One of the fundamental ideas in dynamical systems is hyperbolicity. One lesson from the study of the dynamics of maps of C is that hyperbolicity does not stand alone as a dynamical property, rather, it is one of a sequence of interesting properties which can be defined in terms of recurrence properties of critical points. These one dimensional properties include the critical finiteness property, semi-hyperbolicity, the Collet-Eckmann property and others. In this paper we introduce a dynamical property of polynomial dif-feomorphisms that generalizes hyperbolicity in the way that semi-hyperbolicity generalizes hyperbolicity for polynomial maps of C. In one dimensional complex dynamics generalizations of hyperbolicity are typically defined in terms of recurrence properties of critical points. Since we are dealing with diffeomorphisms of C 2 there are no critical points, and we must use other methods. One way to approach expansion properties is via a certain canonical metric on unstable tangent spaces of periodic saddle points that we define. A mapping is said to the quasi-expanding if this metric is uniformly expanded. Although this metric is canonical it need not be equivalent to the Euclidean metric. It follows that quasi-expansion need not correspond to uniform expansion in the usual sense. We will see in fact that quasi-expansion is strictly weaker than uniform expansion. If both f and f −1 are quasi-expanding we say that f is quasi-hyperbolic. We will show in this paper that quasi-hyperbolic diffeomorphisms have a great deal of interesting structure. Using this structure we develop criteria for showing that certain quasi-expanding diffeomorphisms are uniformly hyperbolic. This criteria for hyperbolicity (as well as the general structure of quasi-hyperbolic diffeomorphisms) plays and important role in the study of real diffeomorphisms of maximal entropy which is carried out in [BS]. We now define the canonical metric on which the definition of quasi-expansion is based. We let S denote …

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تاریخ انتشار 2001