Canonical models on strongly convex domains via the squeezing function

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