Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-19T23:21:33.262Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Lagrange and Markov dynamical spectra

Published online by Cambridge University Press:  11 April 2016

SERGIO AUGUSTO ROMAÑA IBARRA  and
CARLOS GUSTAVO T. DE A. MOREIRA
SERGIO AUGUSTO ROMAÑA IBARRA
Affiliation:
Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Centro de Tecnologia - Bloco C - Cidade Universitária Ilha do Fundão, cep 21941-909, Rio de Janeiro, Brasil email sergiori@im.ufrj.br
CARLOS GUSTAVO T. DE A. MOREIRA
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, cep 22460-320, Rio de Janeiro, Brasil email gugu@impa.br
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the Lagrange and the Markov dynamical spectra associated to horseshoes on a surface with Hausdorff dimension greater than one. We show that for a ‘large’ set of real functions on the surface and for ‘typical’ horseshoes with Hausdorff dimension greater than one, both the Lagrange and the Markov dynamical spectra have persistently non-empty interior.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 5 , August 2017 , pp. 1570 - 1591
Copyright
© Cambridge University Press, 2016 

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

Cusick, T. W. and Flahive, M. E.. The Markoff and Lagrange Spectra (Math Surveys and Monographs, 30) . American Mathematical Society, Providence, RI, 1989.Google Scholar
Freiman, G. A.. Diophantine aproximation and the geometry of numbers (Markov’s problem). Kalinin. Gosudarstv. Univ. Kalink (1975).Google Scholar
Hall, M.. On the sum and product of continued fractions. Ann. of Math. (2) 48 (1947), 993–996.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54) . Cambridge University Press, Cambridge, 1995.Google Scholar
Moreira, C. G.. Conjuntos de Cantor, Dinâmica e Aritmética (22°Colóquio Brasileiro de Matemática) . IMPA, Rio de Janeiro, 1999.Google Scholar
Moreira, C. G. and Yoccoz, J.-C.. Some fundamental properties of plane sets of fractional dimension. Ann. of Math. (2) 154 (2001), 45–96.Google Scholar
Moreira, C. G. and Yoccoz, J.-C.. Tangencies homoclines stables pour des ensembles hyperboliques de grande dimension fractale. Ann. Sci. Éc. Norm. Supér. 43(4) (2010), 1–68.Google Scholar
Palis, J. and Takens, F.. Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations (Fractal Dimensions and Infinitely Many Atractors) (Cambridge Studies in Advanced Mathematics, 35) . Cambridge University Press, Cambridge, 1993.Google Scholar
Shub, M.. Global Stability of Dynamical Systems (with the collabaration of Albert Fathi and Rémi Langevin) . Springer, New York, 1987.Google Scholar
Sullivan, D.. Differentiable structures on fractal-like sets, determined by intrinsic scaling functions on dual cantor sets. The Mathematical Heritage of Herman Weyl (Durham, NC, 1987) (Proceedings of Symposia in Pure Mathematics) . American Mathematical Society, Providence, RI, 1988, 48(4) (1987), 15–23.Google Scholar