Abstract
We consider Hausdorff dimension and limit capacity of basic sets (horse - shoes) of C1 two-dimensional diffeomorphisms and show that they depend continuously on the diffeomorphism. For the restriction of the horseshoe to a stable (unstable) manifold, the result had been proved by McCluskey and Manning using the thermodyna - mic formalism. Our proof simply makes use of Hölder conjugancies between nearby horseshoes and Hölder stable and unstable foliations with Hölder exponents close to one. As a consequence, the local Hausdorff dimension and limit capacity at any point of the horseshoe are equal and independent of the point. Also, for an open and dense subset of C1 surface diffeomorphisms their horseshoes have Hausdorff dimension smaller than two.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Bowen., A horseshoe with positive measure, Inventiones Math. 29 (1975), 203–204.
R. Bowen., Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, vol. 470, Springer Verlag, 1975.
M. Hirsch and C. Pubh., Stable manifolds and hyperbolic sets, Proc. Symp. in Pure Math. 14 (1970), 133–163.
R. Mañé., The Hausdorff dimension of horseshoe of diffeomorphisms of surfaces, IMPA's preprint (1987).
M. McCluskey and A. Manning., Hausdorff dimension for horseshoe, Ergod. Th. and Dynam. Syst. 3 (1983), 231–260.
J.M. Marstrand., The dimension of cartesian product sets, Proc. Cambridge Phil. Soc. 50 (1954), 198–202.
J. Palis and F. Takens., Hyperbilicity and the creation of homoclinic orbits, Annals of Math. 118 (1983), 383–421.
J. Palis and F. Takens., Cycles and measure of bifurcation sets for two dimensional diffeomorphisms, Inventiones Math. 82 (1985), 397–422.
J. Palis and F. Takens., Homoclinic bifurcations and hyperbolic dynamics, Notes of XVI Braz. Math. Colloquium, IMPA (1987).
D. Ruelle., Repellors for real analytic maps, Ergod. Th. and Dynam. Syst. 2 (1982), 99–107.
F. Takens., Limit capacity and Hausdorff dimension of dynamically defined Cantor sets, IMPA's preprint (1987).
Editor information
Rights and permissions
Copyright information
© 1988 Springer-Verlag
About this paper
Cite this paper
Palis, J., Viana, M. (1988). On the continuity of Hausdorff dimension and limit capacity for horseshoes. In: Bamón, R., Labarca, R., Palis, J. (eds) Dynamical Systems Valparaiso 1986. Lecture Notes in Mathematics, vol 1331. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083071
Download citation
DOI: https://doi.org/10.1007/BFb0083071
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50016-2
Online ISBN: 978-3-540-45889-0
eBook Packages: Springer Book Archive