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