Advanced Search
Article Contents
Article Contents

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

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

Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • 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.

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


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Forward coray

  • [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. ArtigianiL. 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. CerqueiraC. 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. HubertL. 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. HasselblattIntroduction 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. TakensHyperbolicity 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. ShiohamaT. Shioya and  M. TakanaThe 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.
  • 加载中



Article Metrics

HTML views(1409) PDF downloads(196) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint