Geometry and combinatorics of Julia sets of real quadratic maps |
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Authors: | M F Barnsley J S Geronimo A N Harrington |
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Institution: | (1) School of Mathematics, Georgia Institute of Technology, 30332 Atlanta, Georgia |
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Abstract: | For real a correspondence is made between the Julia setB
forz (z– )2, in the hyperbolic case, and the set of -chains ± ( ± ( ±..., with the aid of Cremer's theorem. It is shown how a number of features ofB can be understood in terms of -chains. The structure ofB
is determined by certain equivalence classes of -chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of -chains. The functional equations obeyed by attractive cycles are investigated, and their relation to -chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets and -chains. Certain Julia sets associated with the Feigenbaum function and some theorems of Lanford are discussed.Supported by NSF grant No. MCS-8104862.Supported by NSF grant No. MCS-8203325. |
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Keywords: | Iterated maps Julia sets cascades of bifurcations Feigenbaum functional equation universal scaling |
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