Canonical Models on Strongly Convex Domains via the Squeezing Function

We prove that if a holomorphic self-map f : Ω → Ω of a bounded strongly convex domain Ω ⊂ C q with smooth boundary is hyperbolic then it admits a natural semi-conjugacy with a hyperbolic automorphism of a possibly lower dimensional ball B k . We also obtain the dual result for a holomorphic self-map f : Ω → Ω with a boundary repelling fixed point. Both results are obtained by rescaling the dynamics of f via the squeezing function.