Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T13:35:58.322Z Has data issue: false hasContentIssue false
  • English
  • Français

Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

Part of: Diophantine approximation, transcendental number theory Riemann surfaces Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  11 February 2019

MAURO ARTIGIANI Open the ORCID record for MAURO ARTIGIANI [Opens in a new window] ,
LUCA MARCHESE  and
CORINNA ULCIGRAI
MAURO ARTIGIANI
Affiliation:
Universidad de los Andes, Cra 1 #18A – 12, Bogotá, 111711, Colombia email [email protected]
LUCA MARCHESE
Affiliation:
Université Paris 13, Sorbonne Paris Cité, LAGA, UMR 7539, 99 Avenue Jean-Baptiste Clément, 93430 Villetaneuse, France email [email protected]
CORINNA ULCIGRAI
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excursion into a cusp both with respect to a natural height function or, more generally, with respect to any proper function. We prove the existence of a Hall ray for the Lagrange spectrum of any non-cocompact, finite covolume Fuchsian group with respect to any given cusp, both when the penetration is measured by a height function induced by the imaginary part as well as by any proper function close to it with respect to the Lipschitz norm. This shows that Hall rays are stable under (Lipschitz) perturbations. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics and produce a geometric continued fraction-like expansion and some of the ideas in Hall’s original argument. A key element in the proof of the results for proper functions is a generalization of Hall’s theorem on the sum of Cantor sets, where we consider functions which are small perturbations in the Lipschitz norm of the sum.

Keywords

Lagrange spectrumHall rayFuchsian groupsboundary expansionssum of Cantor sets

MSC classification

Primary: 11J06: Markov and Lagrange spectra and generalizations 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 30F35: Fuchsian groups and automorphic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 8 , August 2020 , pp. 2017 - 2072
Check for updates
Copyright
© Cambridge University Press, 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Artigiani, M., Marchese, L. and Ulcigrai, C.. The Lagrange spectrum of a Veech surface has a Hall ray. Groups Geom. Dyn. 10 (2016), 1287–1337.CrossRefGoogle Scholar
Astels, S.. Cantor sets and numbers with restricted partial quotients. Trans. Amer. Math. Soc. 352 (2000), 133–170.CrossRefGoogle Scholar
Beardon, A.. The Geometry of Discrete Groups (Graduate Texts in Mathematics, 91). Springer, New York, 1995.Google Scholar
Boshernitzan, M.. A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J. 52 (1985), 723–752.CrossRefGoogle Scholar
Boshernitzan, M. and Delecroix, V.. From a packing problem to quantitative recurrence in [0, 1] and the Lagrange spectrum of interval exchanges. Discrete Anal. 10 (2017), 1–25.Google Scholar
Bowen, R. and Series, C.. Markov maps associated with Fuchsian groups. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 153–170.CrossRefGoogle Scholar
Cerqueira, A., Matheus, C. and Moreira, C. G.. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. J. Mod. Dyn. 12 (2018), 151–174.CrossRefGoogle Scholar
Cusick, T. W. and Flahive, M. E.. The Markoff and Lagrange Spectra (Mathematical Surveys and Monographs, 30). American Mathematical Society, Providence, RI, USA, 1989.CrossRefGoogle Scholar
Edmonds, A., Ewing, J. and Kulkarni, R.. Torsion free subgroups of Fuchsian groups and tessellations of surfaces. Invent. Math. 69 (1982), 331–346.CrossRefGoogle Scholar
Ferenczi, S.. Dynamical generalizations of the Lagrange spectrum. J. Anal. Math. 118 (2012), 19–53.CrossRefGoogle Scholar
Ford, L.. Fractions. Amer. Math. Monthly 45 (1938), 586–601.CrossRefGoogle Scholar
Fox, R.. On Fenchel’s conjecture about F-groups. Matematisk Tidsskrift. B (1952), 61–65.Google Scholar
Freiman, G. A.. Diophantine Approximation and Geometry of Numbers (The Markoff spectrum). Kalininskii Gosudarstvennyi Universitet, Moscow, 1973.Google Scholar
Haas, A.. Diophantine approximation on hyperbolic Riemann surfaces. Acta Math. 156 (1986), 33–82.CrossRefGoogle Scholar
Haas, A. and Series, C.. The Hurwitz constant and Diophantine approximation on Hecke groups. J. Lond. Math. Soc. (2) 34 (1986), 219–334.CrossRefGoogle Scholar
Hall, M.. On the sum and products of continued fractions. Ann. of Math. (2) 48 (1947), 966–993.CrossRefGoogle Scholar
Hersonsky, S. and Paulin, F.. Diophantine approximation for negatively curved manifolds. Math. Z. 241 (2002), 181–226.CrossRefGoogle Scholar
Hersonsky, S. and Paulin, F.. On the almost sure spiraling of geodesics in negatively curved manifolds. J. Differential Geom. 85 (2010), 271–314.CrossRefGoogle Scholar
Hubert, P., Lelièvre, S., Marchese, L. and Ulcigrai, C.. The Lagrange spectrum of some square-tiled surfaces. Israel J. Math. 225 (2018), 553–607.CrossRefGoogle Scholar
Hubert, P., Marchese, L. and Ulcigrai, C.. Lagrange spectra in Teichmüller dynamics via renormalization. Geom. Funct. Anal. 25 (2015), 180–255.CrossRefGoogle Scholar
Hurwitz, A.. Über die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche. Math. Ann. 39 (1891), 279–284.CrossRefGoogle Scholar
Katok, S.. Fuchsian Groups (Chicago Lectures in Mathematics). University of Chicago Press, Chicago, IL, 1992.Google Scholar
Lehner, J.. A diophantine property of the Fuchsian groups. Pacific J. Math. 2 (1952), 327–333.CrossRefGoogle Scholar
Lehner, J.. A Short Course in Automorphic Functions. Holt, Rinehart and Winston, New York, 1966.Google Scholar
Markoff, A.. Sur les formes quadratiques binaires indéfinies. Math. Ann. 17 (1880), 379–399.CrossRefGoogle Scholar
Matheus, C.. The Lagrange and Markov spectra from the dynamical point of view. Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, CIRM Jean-Morlet Chair Subseries, Fall 2016 (Lecture Notes in Mathematics, 2213). Springer, Berlin, 2018, pp. 259–291.Google Scholar
Maucourant, F.. Sur les spectres de Lagrange et de Markoff des corps imaginaires quadratiques. Ergod. Th. & Dynam. Sys. 23 (2003), 193–205.CrossRefGoogle Scholar
Moreira, C. G.. Geometric properties of the Markov and Lagrange spectra. Ann. of Math. (2) 188(1) (2018), 145–170.CrossRefGoogle Scholar
Moreira, C. G. and Yoccoz, J.C.. Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2) 54 (2001), 45–96.CrossRefGoogle Scholar
Newhouse, S. E.. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 101–151.CrossRefGoogle Scholar
Parkkonen, J. and Paulin, F.. Diophantine approximation on hyperbolic surfaces. Rigidity in Dynamics and Geometry. Eds. Burger, M. and Iozzi, A.. Springer, Berlin, 2002, Appendix to ‘Diophantine approximation in negatively curved manifolds and in the Heisenberg group’.Google Scholar
Parkkonen, J. and Paulin, F.. Prescribing the behaviour of geodesics in negative curvature. Geom. Topol. 14 (2010), 277–392.CrossRefGoogle Scholar
Perron, O.. Die Lehre von den Kettenbrüchen. Chelsea, New York, 1950.Google Scholar
Ratcliffe, J. G.. Foundations of Hyperbolic Manifolds (Graduate Texts in Mathematics, 149). Springer, New York, 2006.Google Scholar
Schmidt, T. A. and Sheingorn, M.. Riemann surfaces have Hall rays at each cusp. Illinois J. Math. 41 (1997), 378–397.CrossRefGoogle Scholar
Series, C.. The modular surface and continued fractions. J. Lond. Math. Soc. (2) 31 (1985), 69–80.CrossRefGoogle Scholar
Series, C.. Geometrical methods of symbolic coding. Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces. Eds. Bedford, T., Keane, M. and Series, C.. Oxford University Press, New York, 1991, pp. 121–151.Google Scholar
Tukia, P.. On discrete groups of the unit disk and their isomorphisms. Ann. Acad. Sci. Fenn. Series A I 504 (1972), 1–45.Google Scholar
Vulakh, L. Ya.. Diophantine approximation in ℝn. Trans. Amer. Math. Soc. 347 (1995), 573–585.Google Scholar
Vulakh, L. Ya.. Diophantine approximation on Bianchi groups. J. Number Theory 54 (1995), 73–80.CrossRefGoogle Scholar
Vulakh, L. Ya.. On Hurwitz constants for Fuchsian groups. Canad. J. Math. 49 (1997), 405–416.CrossRefGoogle Scholar
Vulakh, L. Ya.. The Markov spectra for triangle groups. J. Number Theory 67 (1997), 11–28.CrossRefGoogle Scholar
Vulakh, L. Ya.. The Markov spectra for Fuchsian groups. Trans. Amer. Math. Soc. 352 (2000), 4067–4094.CrossRefGoogle Scholar