Citation/Abstract

We study repellers for regular polynomial endomorphisms of the space [Special characters omitted.] . As our main result we prove that if the forward critical orbit of such a map f is disjoint from the set K of all points with bounded forward f -orbits, then f is expanding on K . In the proof we use hyperbolic metrics on neighborhoods of K given by the Green function. We also prove that f is semiconjugate on K to a subshift of finite type and that K has Lebesgue measure zero. Under additional assumptions we construct a topological conjugacy between f and the d ^k -shift (where d ≥ 2 is the algebraic degree of f ), which also yields an isomorphism of Bernoulli systems, and prove that K is a mixing repeller for f . We give a survey of known one- and multidimensional examples of complex polynomial maps that are expanding on certain invariant sets. We provide necessary background in several complex variables, hyperbolic geometry and dynamics of non-invertible maps.