Let X be a quasi-projective variety and f : X → X a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical degree d 1 (f ) of f equals the arithmetic degree α f (P ) at a point P having Zariski dense f -forward orbit. Assuming X is a smooth affine surface, such that the log Kodaira dimension κ(X) is non-negative (resp. the étale fundamental group π ét 1 (X) is infinite), we confirm AZO, (hence) ZDO, and KSC (when deg(f ) ≥ 2) (resp. AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every surjective endomorphism on any projective variety with "larger" first dynamical degree (resp. every dominant endomorphism of any semiabelian variety).
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