Dynamical studies in several complex variables

This paper deals with different aspects of dynamical systems in several complex variables. It contains the following six papers.

I. Hyperbolic dynamics of endomorphisms. We provide a written account of semilocal and global results for hyperbolic dynamics of endomorphisms.

II. Holomorphic motions of hyperbolic sets (submitted for publication). We study how hyperbolic sets of holomorphic automorphisms and endomorphisms vary under holomorphic perturbations of the map.

III. Some properties of 2-critically finite holomorphic maps of P$\sp2$ (to appear in Ergodic Theory Dynam. Systems). We sharpen previous results by Fornaess and Sibony and by Ueda, by showing that repelling periodic points, as well as the preimages of any given point, are dense in P$\sp2$ for a 2-critically finite map.

IV. Dynamics of polynomial skew products on C$\sp2$: exponents, connectedness and expansion. Polynomial skew products on C$\sp2$ are holomorphic maps of P$\sp2$ whose dynamics resemble that of a one-dimensional polynomial. We study the relation between the critical set, connectedness of Julia sets, Lyapunov exponents, and expansion.

V. Sums of Lyapunov exponents for some polynomial maps of C$\sp2$ (accepted by Ergodic Theory Dynam. Systems). Using a laminar structure for the invariant current, we prove a formula for the sum of the Lyapunov exponents of some polynomial maps of C$\sp2$ with respect to an invariant measure of maximal entropy.

VI. Regular polynomial endomorphisms of C$\sp{k}$ (with E. Bedford). We study the dynamics of polynomial endomorphisms of C$\sp{k}$ that extend holomorphically to P$\sp{k}$; these are called regular. Using techniques from pluripotential theory and hyperbolic dynamics we prove results analogous to those for polynomial mappings of C.