  In this paper we prove the following theorem. Let $f$ be a dominant
endomorphism of a smooth projective surface over an algebraically closed field
of characteristic $0$. If there is no nonconstant invariant rational function
under $f$, then there exists a closed point whose orbit under $f$ is Zariski
dense. This result gives us a positive answer to the Zariski dense orbit
conjecture proposed by Medvedev and Scanlon, by Amerik, Bogomolov and Rovinsky,
and by Zhang, for endomorphisms of smooth projective surfaces.
  Moreover, we define a new canonical topology on varieties over an
algebraically closed field which has finite transcendence degree over
$\mathbb{Q}$. We call it the adelic topology. The adelic topology is stronger
than the Zariski topology and an irreducible variety is still irreducible in
this topology. Using the adelic topology, we propose an adelic verison of the
Zariski dense orbit conjecture. This version is stronger then the original one
and it quantifies how many such orbits there are. We also proved this adelic
version for endomorphisms of smooth projective surfaces. Moreover, we proved
the adelic verison of the Zariski dense orbit conjecture for endomorphisms of
abelian varieties and split polynomial maps. This yields new proofs for the
original version in this two cases.
  In Appendix A, we study the endomorphisms on the $k$-affinoid spaces. We show
that for certain endomorphism $f$ on a $k$-affinoid space $X$, the attractor
$Y$ of $f$ is a Zariski closed subset and the dynamics of $f$ semi-conjugates
to its restriction on $Y.$ A special case of this result is used in the proof
of the main theorem.
  In Appendix B, written in collaboration with Thomas Tucker, we prove the
Zariski dense orbit conjecture for endomorphisms of $(\mathbb{P}^1)^N.$
