On the invariant sets of a family of quadratic maps |
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Authors: | Barnsley M F Geronimo J S Harrington A N |
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Institution: | (1) School of Mathematics, Georgia Institute of Technology, 30332 Atlanta, GA, USA |
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Abstract: | The Julia setB
for the mappingz (z–)2 is considered, where is a complex parameter. For 2 a new upper bound for the Hausdorff dimension is given, and the monic polynomials orthogonal with respect to the equilibrium measure onB
are introduced. A method for calculating all of the polynomials is provided, and certain identities which obtain among coefficients of the three-term recurrence relations are given. A unifying theme is the relationship betweenB
and -chains ± (± (± ...), which is explored for –1/42 and for with ||1/4, with the aid of the Böttcher equation. ThenB
is shown to be a Hölder continuous curve for ||<1/4.Supported by NSF Grant MCS-8104862Supported by NSF Grant MCS-8002731 |
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