The Closing Lemma for Riemann Surfaces
Date
2023
Authors
Nguyen, Wills Ton Minh
Editors
Advisors
Lárusson, Finnur
Baraglia, David
Baraglia, David
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Thesis
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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.
School/Discipline
School of Computer and Mathematical Sciences
Dissertation Note
Thesis (M.Phil.) -- University of Adelaide, School of Computer and Mathematical Sciences, 2023
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