Variations of Hausdorff Dimension in the Exponential Family

2010/03/26 by Guillaume Havard, Havard, Guillaume, Mariusz Urbanski +3
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.1003.5174

32 pages. A paraître dans Annales Academiæ Scientiarum Fennicæ Mathematica

arxiv created 2010/03/26 · arxiv updated 2010/03/29

Abstract

In this paper we deal with the following family of exponential maps (f_λ:z↦ λ(e^z-1))_{λ∈ [1,+∞)}. Denoting d(λ) the hyperbolic dimension of f_λ. It is known that the function λ↦ d(λ) is real analytic in (1,+∞), and that it is continuous in [1,+∞). In this paper we prove that this map is C¹ on [1,+∞), with d'(1⁺)=0. Moreover, depending on the value of d(1), we give estimates of the speed of convergence towards 0.

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