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Let $X$ be a quasi-projective variety and $f\colon X\to X$ a finitesurjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), andAdelic Zariski Dense Orbit Conjecture (AZO). We consider alsoKawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamicaldegree $d_1(f)$ of $f$ equals the arithmetic degree $\alpha_f(P)$ at a point$P$ having Zariski dense $f$-forward orbit. Assuming $X$ is a smooth affinesurface, such that the log Kodaira dimension $\bar{\kappa}(X)$ is non-negative(resp. the \'etale fundamental group $\pi_1^{\text{\'et}}(X)$ is infinite), weconfirm AZO, (hence) ZDO, and KSC (when $\operatorname{deg}(f)\geq 2$) (resp.AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for everysurjective endomorphism on any projective variety with ''larger'' firstdynamical degree (resp. every dominant endomorphism of any semiabelianvariety).

Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures