The Closing Lemma for Riemann Surfaces

Abstract

The closing lemma is a result in dynamical system theory originating from the study of orbits of celestial bodies. In general, it refers to the problem of perturbing a dynamical system so as to obtain an arbitrarily close system for which there is a periodic orbit passing through a given point with a recurrence property. The problem often takes a variety of forms depending on the constraints one imposes and the setting of the given dynamical system, with many closing lemmas still unproven today. In this thesis, we prove closing lemmas in the setting of Riemann surfaces with dynamical systems determined by holomorphic endomorphisms, and with points given the non-wandering property. We aim to provide elementary proofs of these results using the techniques and powerful machinery available to us from Riemann surface theory and the theory of holomorphic dynamics in one complex variable, amongst other areas. Detailed proofs that the closing lemma holds for endomorphisms of the plane C, punctured plane C∗, complex tori, and all Riemann surfaces of hyperbolic type will be presented, with the former two cases forming the main body of the thesis. For the case of the Riemann sphere P, we furnish a proof that the closing lemma holds provided that the given endomorphism admits no Siegel discs and Herman rings.