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In this paper, we prove that if an area-preserving non-degeneratediffeomorphism on the open disk which extend smoothly to the boundary withnon-degeneracy has at least 2 interior periodic points, then there areinfinitely many positive hyperbolic periodic points on the interior. As anapplication, we prove that if a non-degenerate universally tight contact3-dimentional lens space has a Birkhoff section of disk type and at least 3simple periodic orbits, there are infinitely many simple positive hyperbolicorbits. In particular, we have that a non-degenerate dynamically convex contact3-sphere has either infinitely many simple positive hyperbolic orbits orexactly two simple elliptic orbits, which gives a refinement of the resultproved by Hofer, Wysocki and Zehnder in \cite{HWZ2} under non-degeneracy.

Area-preserving diffeomorphisms on the disk and positive hyperbolic orbits