Topological Entropy And Growth Rate Of Periodic Points For Endomorphisms

Due to the complexity of iteration in dynamical systems,Alder,Konheim and McAndrew proposed the concept.of topological entropy to describe the complexity of systems.In uniformly hyperbolic systems.topological entropy is often related to the growth rate of periodic orbits.The relationship between topological entropy and growth rate of periodic orbits for endomorphisms is important for the discussion of complexity of dynamical systems.The previous works are all about diffeomorphisms,in this paper we mainly establish the relationship between topological entropy and growth rate of periodic orbits for endomorphisms.We mainly use the shadowing lemma of endomorphisms version to estimate the growth rate of periodic orbits.For a endomorphism f on a compact Riemannian manifold.Pnδ(f)is defined as the set of n-periodic points with Lyapunov exponents δaway from zero.We mainly consider two cases:(1)f is a C1 endomorphism of S1;(2)f is a C1 partially hyperbolic endomorphism of two-dimensional compact.Riemannian manifold M.In both cases.if htop(f)>0.for any δ∈(0.htop(f)).we have(?).