Mathematica
Volumen 36, 2011, 165-175

# DYNAMICS OF A HIGHER DIMENSIONAL ANALOG OF THE TRIGONOMETRIC FUNCTIONS

## Walter Bergweiler and Alexandre Eremenko

Christian-Albrechts-Universität zu Kiel, Mathematisches Seminar
Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany; bergweiler 'at' math.uni-kiel.de

Purdue University, Department of Mathematics
West Lafayette, IN 47907-2067, U.S.A.; eremenko 'at' math.purdue.edu

Abstract. We introduce a quasiregular analog F of the sine and cosine function such that, for a sufficiently large constant \lambda, the map x \mapsto \lambda F(x) is locally expanding. We show that the dynamics of this map define a representation of Rd, d \geq 2, as a union of simple curves \gamma : [0,\infty) \to Rd which tend to \infty and whose interiors \gamma* = \gamma((0,\infty)) are disjoint such that the union of all \gamma* has Hausdorff dimension 1.

2000 Mathematics Subject Classification: Primary 30D05, 30C65, 37F10, 37F35.

Key words: Dynamics of entire functions, quasiregular map, Zorich map, Julia set, escaping set, Devaney hair, Hausdorff dimension.

Reference to this article: W. Bergweiler and A. Eremenko: Dynamics of a higher dimensional analog of the trigonometric functions. Ann. Acad. Sci. Fenn. Math. 36 (2011), 165-175.

doi:10.5186/aasfm.2011.3610