On the Estimation of Topological Entropy on Surfaces

We provide a rigorous lower bound for the topological entropy of planar diffeomorphisms in terms of the geometry of finite pieces of stable and unstable manifolds of hyperbolic periodic points. Our results suggest the possibility of writing computer programs which would automate the estimation of reasonable approximations for the topological entropy of mappings and differential equations. Applying them to the standard Henon map H(x,y) = (1+ y−ax2,bx) with a = 1.4, b = 0.3 gives the lower bound htop(H)≥ 0.46469.