Polynomial diffeomorphisms of C2. VIII: Quasi-expansion

We introduce the notion of quasi-expansion in the context of polynomial diffeomorphisms of C2. Like hyperbolic diffeomorphisms, quasi-expanding maps have uniformly large multipliers at saddle points. On the other hand, unlike the hyperbolic case, quasi-expanding maps can have tangencies between stable and unstable manifolds. We characterize quasi-expansion in a number of ways and develop some of the structure they possess. Quasi-expansion was motivated by the study of real polynomial diffeomorphisms of maximal entropy, and our study of maximal entropy diffeomorphisms relies on the results of this paper.

[1]  Laminations in holomorphic dynamics , 1994, math/9412233.

[2]  J. Milnor,et al.  Dynamics in One Complex Variable: Introductory Lectures , 2000 .

[3]  P. Hjorth,et al.  Real and complex dynamical systems , 1995 .

[4]  Kaushal Verma.  Boundary regularity of correspondences in ${\bf C^2}$ , 1999 .

[5]  Eric Bedford,et al.  Polynomial diffeomorphisms of $\bf{C}^2$: VI. Connectivity of $J$ , 1996 .

[6]  J. M. Thomas,et al.  Conformal Invariants. , 1926, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Eric Bedford,et al.  Polynomial diffeomorphisms of C2: currents, equilibrium measure and hyperbolicity , 1991 .

[8]  Eric Bedford,et al.  Real polynomial diffeomorphisms with maximal entropy: Tangencies , 2001 .

[9]  Eric Bedford,et al.  Polynomial diffeomorphisms of C2: VII. Hyperbolicity and external rays , 1999 .

[10]  Wolfgang H. J. Fuchs,et al.  The Theory of Functions of One Complex Variable , 1967 .

[11]  Polynomial diffeomorphisms ofC2. IV: The measure of maximal entropy and laminar currents , 1992, math/9205210.

[12]  F. Ledrappier,et al.  A proof of the estimation from below in Pesin's entropy formula , 1982, Ergodic Theory and Dynamical Systems.

[13]  Ralph W. Oberste-Vorth,et al.  Henon mappings in the complex domain II: projective and inductive limits of polynomials , 1994 .

[14]  Gert Sabidussi,et al.  Complex Potential Theory , 1994 .

[15]  N. Sibony,et al.  Complex dynamics in higher dimensions , 1994 .

[16]  J.-M. Carette.  Liens entre la géométrie et la dynamique des ensembles de Julia , 1997 .

[17]  J. Smillie,et al.  Polynomial diffeomorphisms of C2. III: Ergodicity, exponents and entropy of the equilibrium measure , 1992 .

[18]  Z. Slodkowski.  Holomorphic motions and polynomial hulls , 1991 .