Variations of Hausdorff Dimension in the Exponential Family

TL;DR

This paper investigates the smoothness and convergence properties of the hyperbolic dimension function in a family of exponential maps, establishing its differentiability at the boundary point and analyzing the convergence rate based on initial conditions.

Contribution

It proves that the hyperbolic dimension function is continuously differentiable on the entire parameter interval and provides estimates for the convergence speed depending on initial dimension values.

Findings

The function d(λ) is C^1 on [1,+∞).

The derivative d'(1^+) equals zero.

Convergence speed estimates depend on d(1).

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