**Title: **Periodic point growth for C^2 maps of the two sphere

**Speaker: **Yun Yang (CUNY)

**Abstract: **There are two basic mechanisms producing periodic orbits in a dynamical systems: contraction (via the Contraction mapping theorem, Banach fixed point theorem) and degree (via topological methods such as the Lefschetz theorem).These two mechanisms play an important role in the joint work of Enrique Pujals, Michael Shub and myself with assistance from Sylvain Crovisier on Shub’s conjecture: Let f : M -> M be a C2 map of a compact manifold. Then the exponential growth rate in fixed points of fn bounded below by the growth rate of the Lefschetz numbers of f^n. In this talk, I will present the proof of this conjecture in the case where f: S2 -> S2 has positive entropy and reverses orientation in the direction of vanishing exponents.