Topology Proceedings
Document # baam-11
Topology Proceedings 28 No. 1 (2004), pp. 99-112 |
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Topological bifurcations
Robert L. Devaney
In complex dynamics, the important object of study is the
Julia set of a given holomorphic function. This set contains all of the
points where the map is chaotic. As parameters change, the Julia set
sometimes undergoes rather remarkable changes in topology. We call these
changes topological bifurcations. In this paper we describe a number of
different topological bifurcations, all of which occur in the family
Fa, b(z) = frac1a+ be-z.
We present bifurcations in which the Julia sets of this family are
transformed abruptly: from a Cantor bouquet to a simple closed curve in
the Riemann sphere; from a Cantor bouquet to a Cantor set; from a Cantor
bouquet to the whole Riemann sphere, including the appearance of
infinitely many indecomposable continua.
Keywords: bifurcations, dynamical systems
Mathematics Subject Classification: 37F45 (37F10 37F20)
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Topology Proceedings,
Volume 28, No. 1 (2004)
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