Birational geometry in the study of dynamics of automorphisms and Brody/Mori/Lang hyperbolicity

We survey our recent papers (some being joint ones) about the relation between the geometry of a compact K"ahler manifold and the existence of automorphisms of positive entropy on it. We also use the language of log minimal model program (LMMP) in biraitonal geometry, but not its more sophisticated technical part. We give applications of LMMP to positivity of log canonical divisor of a Mori / Brody / Lang hyperbolic (quasi-) projective (singular) variety.