Variations of Hausdorff Dimension in the Exponential Family

Details Variations of Hausdorff Dimension in the Exponential Family Variations of Hausdorff Dimension in the Exponential Family

2010-03-26

arxiv.org/abs/1003.5174v1

Mathematics

Dynamical Systems

32 pages. A para\^itre dans Annales Academi{\ae} Scientiarum Fennic{\ae} Mathematica

Scientific paper

In this paper we deal with the following family of exponential maps $(f_\lambda:z\mapsto \lambda(e^z-1))_{\lambda\in [1,+\infty)}$. Denoting $d(\lambda)$ the hyperbolic dimension of $f_\lambda$. It is known that the function $\lambda\mapsto d(\lambda)$ 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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