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Critical saddle-node cycles: Hausdorff dimension and persistence of tangencies

Published online by Cambridge University Press:  06 August 2002

LORENZO J. DÍAZ  and
RAUL URES
LORENZO J. DÍAZ
Affiliation:
Departamento Matemática, PUC-Rio, Marquês de S. Vicente 225, 22453-900 Rio de Janeiro, RJ, Brazil (e-mail: lodiaz@mat.puc-rio.br)
RAUL URES
Affiliation:
CC30 IMERL, Facultad de Ingeniería, Universidad de la República, Uruguay (e-mail: ures@fing.edu.uy)
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Abstract

We consider the collapse of a saddle of a horseshoe and a sink via a critical saddle-node bifurcation. In this way one obtains a saddle-node horseshoe. We prove that there is an open set of arcs of diffeomorphisms \{f_\mu\}_{\mu \in I} unfolding generically, say at \mu_0, a saddle-node horseshoe with Hausdorff dimension arbitrarily close to 1/2 so that there is an interval (\mu_0, \mu_0+\varepsilon) in the parameter line such that every diffeomorphism f_\mu, \mu\in (\mu_0, \mu_0+\delta), has a tangency.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 22 , Issue 4 , August 2002 , pp. 1117 - 1140
Copyright
© 2002 Cambridge University Press

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