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Non-expansive derived horseshoes

Part of: Connections with other structures, applications Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  09 April 2009

Kazuhiro Sakai
Kazuhiro Sakai
Affiliation:
Department of Mathematics, Kanagawa University, Yokohama 221-8686, Japan e-mail: kazsaka@cc.kanagawa-u.ac.jp
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Abstract

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For a two parameter family of C3 diffeomorphisms having a homoclinic orbit of tangency derived from a horseshoe, the relationship between the measure of the parameter values at which the diffeomorphism (restricted to a certain compact invariant set containing the horseshoe) is not expansive and the Hausdorff dimension of the horseshoe associated to the homoclinic orbit of tangency is investigated. This is a simple application of the Newhouse-Palis-Takens-Yoccoz theory.

Keywords

expansiveHénon maphorseshoeHausdorff dimensionmeasurethicknesshomoclinic orbit of tangency

MSC classification

Secondary: 54H20: Topological dynamics 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 3 , December 1998 , pp. 405 - 415
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Milnor, J., ‘Non-expansive Hénon maps’, Adv. Math. 69 (1988), 109–114.CrossRefGoogle Scholar
[2]Newhouse, S., ‘The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms’, Publ. Math. IHES 50 (1979), 101–151.CrossRefGoogle Scholar
[3]Palis, J. and Takens, F., ‘Hyperbolicity and the creation of homoclinic orbits’, Ann. Math. 125 (1987), 337–374.CrossRefGoogle Scholar
[4]Palis, J. and Takens, F.Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations (Cambridge Univ. Press, London, 1993).Google Scholar
[5]Palis, J. and Yoccoz, J. C., ‘Homoclinic tangencies for hyperbolic sets of large Hausdorff dimensions’, Acta Math. 172 (1994), 91–136.CrossRefGoogle Scholar
[6]Robinson, C., ‘Bifurcations to infinitely many sinks’, Comm. Math. Phys. 90 (1983), 433–459.CrossRefGoogle Scholar
[7]Sannami, A., ‘An example of a regular Cantor set whose differ ence set is a Cantor set with positive measure’, Hokkaido Math. J. 21 (1992), 7–24.CrossRefGoogle Scholar
[8]Wang, X. J., ‘The Newhouse set has a positive Hausdorff dimension’, Comm. Math. Phys. 131 (1990), 317–332.CrossRefGoogle Scholar