Variations of Hausdorff Dimension in the Exponential Family

Abstract

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 $d^{\prime }(1^{+})=0$. Moreover, depending on the value of $d(1)$, we give estimates of the speed of convergence towards 0.

arXiv comment32 pages. A paraître dans Annales Academiæ Scientiarum Fennicæ Mathematica