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Tying hairs for structurally stable exponentials

Published online by Cambridge University Press:  01 December 2000

RANJIT BHATTACHARJEE
Affiliation:
Department of Mathematics, Boston University, Boston MA 02215, USA (e-mail: bob@bu.edu)
ROBERT L. DEVANEY
Affiliation:
Department of Mathematics, Boston University, Boston MA 02215, USA (e-mail: bob@bu.edu)

Abstract

Our goal in this paper is to describe the structure of the Julia set of complex exponential functions that possess an attracting cycle. When the cycle is a fixed point, it is known that the Julia set is a ‘Cantor bouquet’, a union of uncountably many distinct curves or ‘hairs’. When the period of the cycle is greater than one, infinitely many of the hairs in the bouquet become pinched or attached together. In this paper, we develop an algorithm to determine which of these hairs are attached. Of crucial importance in this construction is the kneading invariant, a sequence that is derived from the topology of the basins of attraction of the attracting cycle.

Type
Research Article
Copyright
© 2000 Cambridge University Press

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