On the Lagrange and Markov dynamical spectra for geodesic flows on surfaces with negative curvature

Carlos Gustavo T. de A. Moreira; Sergio Augusto Romaña Ibarra

doi:10.3934/jmd.2023005

Article Contents

2023, Volume 19: 187-236. Doi: 10.3934/jmd.2023005

This volume Previous Article Some asymptotic properties of random walks on homogeneous spaces Next Article Regularizations of pseudo-automorphisms with positive algebraic entropy

On the Lagrange and Markov dynamical spectra for geodesic flows on surfaces with negative curvature

1.
SUSTech International Center for Mathematics, Shenzhen, Guangdong, People's Republic of China
2.
Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, cep 22460-320, Rio de Janeiro, Brasil
3.
Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Instituto de Matématica, Centro de Tecnologia-Bloco C, Cidade Universitária Ilha do Fundão, cep 21941-909, Rio de Janeiro, Brasil

Received: June 17, 2021

Revised: October 25, 2022

CGTdAM: Partially supported by CNPq, the Palis Balzan Prize and FAPERJ.
SARI: Partially supported by CNPq, Capes, the Palis Balzan Prize and FAPERJ

Abstract / Introduction Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

We consider the Lagrange and the Markov dynamical spectra associated with the geodesic flow on surfaces of negative curvature. We show that for a large set of real functions on the unit tangent bundle and typical metrics with negative curvature and finite volume, both the Lagrange and the Markov dynamical spectra have non-empty interiors.

Keywords:

Mathematics Subject Classification: Primary: 37D40; Secondary: 53C20.

Citation:

Full Text(HTML)

Figure 1. Forward coray

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	D. Anosov, Geodesic Flow on Compact Manifolds of Negative Curvature, Proc. Steklov Math. Inst. (1967), translated, Amer. Math. Soc., Providence, 1969.
[2]	M. Artigiani, L. Marchese and C. Ulcigrai, The Lagrange spectrum of a Veech surface has a Hall ray, Groups Geom. Dyn., 10 (2016), 1287–1337. doi: 10.4171/GGD/384.
[3]	M. Artigiani, L. Marchese and C. Ulcigrai, Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces, Ergodic Theory Dynam. Systems, 40 (2020), 2017-2072. doi: 10.1017/etds.2018.143.
[4]	W. Ballmann, Lectures on Spaces of Nonpositive Curvature, DMV Seminar Band 25, Birkhäuser, 1995. doi: 10.1007/978-3-0348-9240-7.
[5]	Y. Bugeaud, Nonarchimedean quadratic Lagrange spectra and continued fractions in power series fields, Math. Z., 276 (2014), 985-999. doi: 10.1007/s00209-013-1230-1.
[6]	C. Camacho and A. L. Neto, Introdução à Teoria das Folheações, Instituto de Matemática Pura e Aplicada, 1977.
[7]	A. Cerqueira, C. Matheus and C. G. Moreira, Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra, Journal of Modern Dynamics, 12 (2018), 151-174. doi: 10.3934/jmd.2018006.
[8]	A. Cerqueira, C. G. Moreira and S. Romaña, Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra II, Ergodic Theory Dynam. Systems, 42 (2022), 1898–1907. doi: 10.1017/etds.2021.18.
[9]	T. W. Cusick and M. E. Flahive, The Markoff and Lagrange Spectra, Math. Surveys and Monographs, 30, Amer. Math. Soc., Providence, 1989. doi: 10.1090/surv/030.
[10]	P. Eberlein, When is geodesic flow of Anosov type?, J. Diff. Geom., 8 (1973), 437-463. doi: 10.4310/jdg/121443180.
[11]	P. Eberlein, Lattices in spaces of nonpositive curvature, Ann. of Math., 111 (1980), 435-476. doi: 10.2307/1971104.
[12]	G. A. Freiman, Diophantine aproximation and the geometry of numbers (Markov problem), Kalinin. Gosudarstv. Univ., Kalinin, 1975.
[13]	M. Gromov, Manifolds of negative curvature, J. Diff. Geom., 13 (1978), 223-230. doi: 10.4310/jdg/1214434487.
[14]	A. Haas, Diophantine approximation on hyperbolic Riemann surfaces, Acta Math., 156 (1986), 33-82. doi: 10.1007/BF02399200.
[15]	A. Haas and C. Series, The Hurwitz constant and Diophantine approximation on Hecke groups, J. London Math. Soc., 34 (1986), 219-234. doi: 10.1112/jlms/s2-34.2.219.
[16]	M. Hall, On the sum and product of continued fractions, Ann. of Math., 48 (1947), 966-993. doi: 10.2307/1969389.
[17]	E. Heintze, Mannigfaltigkeiten Negativer Krümmung, Universität Bonn, Mathematisches Institut, Bonn, 2002.
[18]	S. Hersonsky and F. Paulin, Diophantine approximation for negatively curved manifolds, Math. Z, 241 (2002), 181-226. doi: 10.1007/s002090200412.
[19]	P. Hubert, L. Marchese and C. Ulcigrai, Lagrange spectra in Teichmüller dynamics via renormalization, Geom. Funct. Anal., 25 (2015), 180-255. doi: 10.1007/s00039-015-0321-z.
[20]	N. Innami, Differentiability of Busemann functions and total excess, Math. Z., 180 (1982), 235-247. doi: 10.1007/BF01318907.
[21]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.
[22]	W. Klingenberg, Riemannian Geometry, Walter de Gruyter, Berlin, 1982.
[23]	W. Klingenberg and F. Takens, Generic properties of geodesic flows, Math. Ann., 197 (1972), 323-334. doi: 10.1007/BF01428204.
[24]	D. Lima, C. Matheus, C. G. Moreira and S. Romaña, Classical and Dynamical Markov and Lagrange Spectra Dynamical, Fractal and Arithmetic Aspects, World Scientific, 2021.
[25]	C. G. Moreira, Geometric properties of the Markov and Lagrange spectra, Ann. of Math., 188 (2018), 145-170. doi: 10.4007/annals.2018.188.1.3.
[26]	C. G. Moreira and J.-C. Yoccoz, Tangencies homoclines stables pour des ensembles hyperboliques de grande dimension fractale, Ann. Sci. Éc. Norm. Supér., 43 (2010), 1-68. doi: 10.24033/asens.2115.
[27]	C. G. T. de A. Moreira and J.-C. Yoccoz, Stable intersections of regular Cantor sets with large Hausdorff dimensions, Ann. of Math., 154 (2001), 45-96. doi: 10.2307/3062110.
[28]	J. Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math., 9 (1956), 673-692. doi: 10.1002/cpa.3160090404.
[29]	Y. Nasu, On asymptotes in a metric space with non-positive curvature, Tohoku Math. J., 9 (1957), 68-95. doi: 10.2748/tmj/1178244920.
[30]	J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, 35, Cambridge University Press, 1993.
[31]	J. Parkkonen and F. Paulin, Prescribing the behavior of geodesic in negative curvature, Geom. Topol., 14 (2010), 277-392. doi: 10.2140/gt.2010.14.277.
[32]	J. Parkkonen and F. Paulin, On the nonarchimedean quadratic Lagrange spectra, Math Z., 294 (2020), 1065-1084. doi: 10.1007/s00209-019-02300-1.
[33]	G. P. Paternain, Geodesic Flows, Progress in Mathematics, 180, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-1600-1.
[34]	S. A. Romaña Ibarra, On the Lagrange and Markov dynamical spectra for Anosov flows in dimesion 3, Qual. Theory Dyn. Syst., 21 (2022), Paper No. 19, 48 pp. doi: 10.1007/s12346-021-00543-0.
[35]	S. A. Romaña Ibarra and C. G. T. de A. Moreira, On the Lagrange and Markov dynamical spectra, Ergodic Theory Dynam. Systems, 37 (2017), 1570-1591. doi: 10.1017/etds.2015.121.
[36]	T. A. Schmidt and M. Sheingorn, Riemann surfaces have Hall rays at each cusp, Illinois J. Math., 41 (1997), 378-397. doi: 10.1215/ijm/1255985734.
[37]	K. Shiohama, T. Shioya and M. Takana, The Geometry of Total Curvature on Complete Open Surfaces, Cambridge Tracts in Mathematics, 159, Cambridge University Press, 2003. doi: 10.1017/CBO9780511543159.
[38]	D. Sullivan, Differentiable structures on fractal-like sets, determined by intrinsic scaling functions on dual Cantor sets, 15–23, Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, 1988. doi: 10.1090/pspum/048/974329.