  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.
