  We define the Kobayashi quotient of a complex variety by identifying points
with vanishing Kobayashi pseudodistance between them and show that if a compact
complex manifold has an automorphism whose order is infinite, then the fibers
of this quotient map are nontrivial. We prove that the Kobayashi quotients
associated to ergodic complex structures on a compact manifold are isomorphic.
We also give a proof of Kobayashi's conjecture on the vanishing of the
pseudodistance for hyperk\"ahler manifolds having Lagrangian fibrations without
multiple fibers in codimension one. For a hyperbolic automorphism of a
hyperk\"ahler manifold, we prove that its cohomology eigenvalues are determined
by its Hodge numbers, compute its dynamical degree and show that its
cohomological trace grows exponentially, giving estimates on the number of its
periodic points.
