Canonical models on strongly convex domains via the squeezing function

We prove that if a holomorphic self-map $f\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to \mathrm{\Omega }$ of a bounded strongly convex domain $\mathrm{\Omega }\subset {\mathbb{C}}^q$ 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 $f\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to \mathrm{\Omega }$ with a boundary repelling fixed point. Both results are obtained by rescaling the dynamics of $f$ via the squeezing function.