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Geometry of the horseshoe $\Lambda$ .
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2018, 12: 151-174.
doi: 10.3934/jmd.2018006
Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra
| 1. | IMPA, Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil |
| 2. | Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), F-93439, Villetaneuse, France |
Let $\varphi_0$ be a smooth area-preserving diffeomorphism of a compact surface $M$ and let $Λ_0$ be a horseshoe of $\varphi_0$ with Hausdorff dimension strictly smaller than one. Given a smooth function $f:M\to \mathbb{R}$ and a small smooth area-preserving perturtabion $\varphi$ of $\varphi_0$, let $L_{\varphi, f}$, resp. $M_{\varphi, f}$ be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of $f$ along the $\varphi$-orbits of points in the horseshoe $Λ$ obtained by hyperbolic continuation of $Λ_0$.
We show that, for generic choices of $\varphi$ and $f$, the Hausdorff dimension of the sets $L_{\varphi, f}\cap (-∞, t)$ vary continuously with $t∈\mathbb{R}$ and, moreover, $M_{\varphi, f}\cap (-∞, t)$ has the same Hausdorff dimension as $L_{\varphi, f}\cap (-∞, t)$ for all $t∈\mathbb{R}$.
Keywords:
Hausdorff dimension,
horseshoes,
Lagrange spectrum,
Markov spectrum,
surface diffeomorphisms.
Mathematics Subject Classification:
Primary: 11J06, 37E30; Secondary: 37D05.
Citation:
Aline Cerqueira, Carlos Matheus, Carlos Gustavo Moreira. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. Journal of Modern Dynamics,
2018, 12: 151-174.
doi: 10.3934/jmd.2018006
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References:
| [1] |
T. Cusick and M. Flahive,
The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs, 30, American Mathematical Society, Providence, RI, 1989.
doi: 10.1090/surv/030. |
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S. Hersonsky and F. Paulin,
Diophantine approximation for negatively curved manifolds, Math. Z., 241 (2002), 181-226.
doi: 10.1007/s002090200412. |
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M. Hirsch, C Pugh and M. Shub,
Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. |
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C. G. Moreira, Geometric properties of the Markov and Lagrange spectra, preprint, 2016, available at arXiv: 1612.05782, accepted for publication in Ann. Math. |
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C. G. Moreira, Geometric properties of images of cartesian products of regular Cantor sets by differentiable real maps, preprint, 2016, available at arXiv: 1611.00933. |
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C. G. Moreira and S. Romaña,
On the Lagrange and Markov dynamical spectra, Ergodic Theory Dynam. Systems, 37 (2017), 1570-1591.
doi: 10.1017/etds.2015.121. |
| [7] |
C. G. Moreira and J.-C. Yoccoz,
Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale, Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 1-68.
doi: 10.24033/asens.2115. |
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J. Palis and F. Takens,
Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, 35 Cambridge University Press, Cambridge, 1993. |
show all references
References:
| [1] |
T. Cusick and M. Flahive,
The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs, 30, American Mathematical Society, Providence, RI, 1989.
doi: 10.1090/surv/030. |
| [2] |
S. Hersonsky and F. Paulin,
Diophantine approximation for negatively curved manifolds, Math. Z., 241 (2002), 181-226.
doi: 10.1007/s002090200412. |
| [3] |
M. Hirsch, C Pugh and M. Shub,
Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. |
| [4] |
C. G. Moreira, Geometric properties of the Markov and Lagrange spectra, preprint, 2016, available at arXiv: 1612.05782, accepted for publication in Ann. Math. |
| [5] |
C. G. Moreira, Geometric properties of images of cartesian products of regular Cantor sets by differentiable real maps, preprint, 2016, available at arXiv: 1611.00933. |
| [6] |
C. G. Moreira and S. Romaña,
On the Lagrange and Markov dynamical spectra, Ergodic Theory Dynam. Systems, 37 (2017), 1570-1591.
doi: 10.1017/etds.2015.121. |
| [7] |
C. G. Moreira and J.-C. Yoccoz,
Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale, Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 1-68.
doi: 10.24033/asens.2115. |
| [8] |
J. Palis and F. Takens,
Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, 35 Cambridge University Press, Cambridge, 1993. |
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