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August  2008, 21(3): 907-928. doi: 10.3934/dcds.2008.21.907

Transversal families of hyperbolic skew-products

Eugen Mihailescu 1, and  Mariusz Urbański 2,

1. 

Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O Box 1-764, RO 014-700, Bucharest
2. 

Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203-1430

Received  June 2007 Revised  January 2008 Published  April 2008

We study families of hyperbolic skew products with the transversality condition and in particular, the Hausdorff dimension of their fibers, by using thermodynamical formalism. The maps we consider can be non-invertible, and the study of their dynamics is influenced greatly by this fact.
    We introduce and employ probability measures (constructed from equilibrium measures on the natural extension), which are supported on the fibers of the skew product. A stronger condition, that of Uniform Transversality is then considered in order to obtain a general formula for Hausdorff dimension of fibers for all base points and almost all parameters.
    In the end we study a large class of examples of transversal hyperbolic families which locally depend linearly on the parameters, and also another class of examples related to complex dynamics.
Keywords: skew products, Hausdorff dimension, hyperbolic maps, transversality..
Mathematics Subject Classification: Primary: 37D35, 37D20; Secondary: 37A3.
Citation: Eugen Mihailescu, Mariusz Urbański. Transversal families of hyperbolic skew-products. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 907-928. doi: 10.3934/dcds.2008.21.907
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