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

Let $X$ be a quasi-projective variety and $fcolon Xto X$ 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 $alpha_f(P)$ at a point $P$ having Zariski dense $f$-forward orbit. Assuming $X$ is a smooth affine surface, such that the log Kodaira dimension $bar{kappa}(X)$ is non-negative (resp. the etale fundamental group $pi_1^{text{et}}(X)$ is infinite), we confirm 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 every surjective endomorphism on any projective variety with larger first dynamical degree (resp. every dominant endomorphism of any semiabelian variety).