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arXiv · 1003.5174

Variations of Hausdorff Dimension in the Exponential Family

Guillaume Havard·Mariusz Urbanski·Michel Zinsmeister
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Abstract

In this paper we deal with the following family of exponential maps $(f_λ:z\mapsto λ(e^z-1))_{λ\in [1,+\infty)}$. Denoting $d(λ)$ the hyperbolic dimension of $f_λ$. It is known that the function $λ\mapsto d(λ)$ is real analytic in $(1,+\infty)$, and that it is continuous in $[1,+\infty)$. In this paper we prove that this map is C$^1$ on $[1,+\infty)$, 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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Guillaume Havard, Mariusz Urbanski, Michel Zinsmeister. 2010-03-26. Variations of Hausdorff Dimension in the Exponential Family. https://arxiv.org/abs/1003.5174

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