Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 35, 2010, 351-378

# VARIATIONS OF HAUSDORFF DIMENSION IN THE EXPONENTIAL FAMILY

## Guillaume Havard, Mariusz Urbanski and Michel Zinsmeister

Université Blaise Pascal, Laboratoire de Mathématiques
63177 Aubière cedex, France; guillaume.havard 'at' math.univ-bpclermont.fr

University of North Texas, Department of Mathematics
P.O. Box 311430, Denton TX 76203-1430, U.S.A.; urbanski 'at' unt.edu

Université d'Orléans, Laboratoire de Mathématiques et Applications
B.P. 6759, 45067 Orléans cedex 2, France; Michel.Zinsmeister 'at' math.cnrs.fr

Abstract. In this paper we deal with the following family of exponential maps (f\lambda : z \mapsto \lambda(ez - 1))\lambda \in [1,+\infty). Denote by d(\lambda) the hyperbolic dimension of f\lambda. It is proved in [Ur,Zd1] that the function \lambda \mapsto d(\lambda) is real-analytic in (1,+\infty), and in [Ur,Zd2] that it is continuous in [1,+\infty). In this paper we prove that this map is C1 on [1,+\infty), with d'(1+) = 0. Moreover we prove that depending on the value of d(1)

d'(1+\varepsilon) \sim -\varepsilon2d(1)-2 if d(1) < 3/2,
|d'(1+\varepsilon)| \lesssim -\varepsilon log\varepsilon if d(1) = 3/2,
|d'(1+\varepsilon)| \lesssim \varepsilon if d(1) > 3/2.

In particular, if d(1) < 3/2, then there exists \lambda0 > 1 such that d(\lambda) < d(1) for any \lambda \in (1,\lambda0).

2000 Mathematics Subject Classification: Primary 37F35, 37F45, 30D05.

Key words: Hausdorff dimension, Julia set, exponential family, parabolic points, thermodynamic formalism, conformal measures.

Reference to this article: G. Havard, M. Urbanski and M. Zinsmeister: Variations of Hausdorff dimension in the exponential family. Ann. Acad. Sci. Fenn. Math. 35 (2010), 351-378.

doi:10.5186/aasfm.2010.3523