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Variation of topological pressure and dimension: from polynomials to complex Hénon maps

Published online by Cambridge University Press:  14 March 2013

CHRISTIAN WOLF
CHRISTIAN WOLF*
Affiliation:
Department of Mathematics, The City College of New York, New York, NY 10031, USA email [email protected]
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Abstract

We study the topological pressure and dimension theory of complex Hénon maps which are small perturbations of one-dimensional polynomials. In particular, we derive regularity results for the generalized pressure function in a neighborhood of the degenerate map (i.e. the polynomial). This unifies results concerning the regularity of the pressure function for polynomials by Ruelle and for complex Hénon maps by Verjovsky and Wu. We then apply this regularity to show that the Hausdorff dimension of the Julia set is a continuous non-differentiable function in a neighborhood of the polynomial. Furthermore, we establish uniqueness of the measure of maximal dimension and show that the Hausdorff dimension of the Julia set of a complex Hénon map is discontinuous at the boundary of the hyperbolicity locus.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 3 , June 2014 , pp. 1018 - 1036
Copyright
Copyright ©2013 Cambridge University Press 

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