A Remark on Papers by Pixton and Oliveira: Genericity of Symplectic Diffeomorphisms of S2 with Positive Topological Entropy

We prove the existence of an open and dense subset of maps f 2 Diff 1 ! (S 2) which have positive topological entropy. It follows that these maps have innnitely many hyperbolic periodic points and an exponential growth rate of hyperbolic periodic points. The proof is an application of Pixton's theorem. Topological entropy characterizes the total exponential orbit complexity of a map with a single number (see KH] for deenitions and properties). Topological entropy, especially in low dimensional cases, provides a wealth of qualitative structural information about the system including the growth rate of the number of periodic orbits K1], existence of large horseshoes K2], and the growth rate of the volume of cells of various dimensions Y]. Any map which possesses a horseshoe, i.e., some power of the map is topologically conjugate to a Bernoulli shift, has positive topological entropy and Katok K1] has shown that the converse is true for surface diieomorphisms. Hence surface diieomorphisms having positive topological entropy exhibit very stochastic behavior on some subset of the surface-possibly a set of Lebesgue measure zero. Thus, the stochastic behavior of a surface diieomorphism with positive topological entropy may not be physically observable. In this note, we observe that there exists an open and dense subset of C 1 symplectic (area-preserving) diieomorphisms (symplectomorphisms) on S 2 having positive topo-logical entropy, and we observe a related result for symplectomorphisms of the 2-torus.