A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature
This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers by
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A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature Jouni Parkkonen and Frédéric Paulin
7.1 Introduction For several decades, tools from dynamical systems, and in particular ergodic theory, have been used to derive arithmetic and number theoretic results in particular Diophantine approximation ones, see for instance the works of Furstenberg, Margulis, Sullivan, Dani, Kleinbock, Clozel, Oh, Ullmo, Lindenstrauss, Einsiedler, Michel, Venkatesh, Marklof, Green-Tao, Elkies-McMullen, Ratner, Mozes, Shah, Gorodnik, Ghosh, Weiss, Hersonsky-Paulin, Parkkonen-Paulin and many others, and the references [Kle2, Lind, Kle1, AMM, Ath, GorN, EiW, PaP5]. In Sect. 7.2.2 of this survey, we introduce a general framework of Diophantine approximation in measured metric spaces, in which most of our arithmetic corollaries are inserted (see the end of Sect. 7.2.2 for references concerning this framework). In order to motivate it, we first recall in Sect. 7.2.1 some very basic and classical results in Diophantine approximation (see for instance [Bug1, Bug2]). A selection (extracted from [PaP1, PaP4, PaP7]) of our arithmetic results are then stated in Sects. 7.2.3–7.2.5, where we indicate how they fit into this framework: Diophantine approximation results (à la Khintchine, Hurwitz, Cusick-Flahive and Farey) of real numbers by quadratic irrational ones, equidistribution of rational points in R (for various height functions), in C and in the Heisenberg group, : : : We will explain in Sect. 7.4.2 the starting point of their proofs, using the geometric and ergodic tools and results previously described in Sect. 7.4.1, where
J. Parkkonen Department of Mathematics and Statistics, P.O. Box 35, 40014 University of Jyväskylä, Finland e-mail: [email protected] F. Paulin () Laboratoire de Mathématique d’Orsay, UMR 8628 Université Paris-Sudet CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France e-mail: [email protected] © Springer International Publishing Switzerland 2017 B. Hasselblatt (ed.), Ergodic Theory and Negative Curvature, Lecture Notes in Mathematics 2164, DOI 10.1007/978-3-319-43059-1_7
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we give an exposition of our work in [PaP6]: an asymptotic formula as t ! C1 for the number of common perpendiculars of length at most t between closed locally convex subsets D and DC in a negatively curved Riemannian orbifold, and an equidistribution result of the initial and terminal tangent vectors v˛ and v˛C of the common perpendiculars ˛ in the outer and inner unit normal bundles of D and DC , respectively. Common perpendiculars have been studied, in various particular cases, sometimes not explicitly, by Basmajian, Bridgeman, Bridgeman-Kahn, Eskin-McMullen, Herrmann, Huber, Kontorovich-Oh, Margulis, Martin-McKeeWambach, Meyerhoff, Mirzakhani, Oh-Shah, Pollicott, Roblin, Shah, the authors and many others (see the comments after Theorem 15 below, and the survey [PaP5] for references). Section 7.3 presents the background notions on the geometry in negative curv
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