  In this paper, we study arithmetic dynamics in arbitrary characteristic, in
particular in positive characteristic. We generalise some basic facts on
arithmetic degree and canonical height in positive characteristic. As
applications, we prove the dynamical Mordell-Lang conjecture for automorphisms
of projective surfaces of positive entropy, the Zariski dense orbit conjecture
for automorphisms of projective surfaces and for endomorphisms of projective
varieties with large first dynamical degree. We also study ergodic theory for
constructible topology. For example, we prove the equidistribution of backward
orbits for finite flat endomorphisms with large topological degree. As
applications, we give a simple proof for weak dynamical Mordell-Lang and prove
a counting result for backward orbits without multiplicities. This gives some
applications for equidistributions on Berkovich spaces.
