Sierpiński and non‐Sierpiński curve Julia sets in families of rational maps | Zendy

Steinmetz Norbert | Zendy

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Sierpiński and non‐Sierpiński curve Julia sets in families of rational maps

Author(s) -

Steinmetz Norbert

Publication year - 2008

Publication title -

journal of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.441

H-Index - 62

eISSN - 1469-7750

pISSN - 0024-6107

DOI - 10.1112/jlms/jdn030

Subject(s) - julia set , cantor set , infinity , mathematics , sierpinski triangle , plane (geometry) , set (abstract data type) , orbit (dynamics) , fractal , pure mathematics , mathematical analysis , geometry , computer science , engineering , programming language , aerospace engineering

We discuss the dynamics as well as the structure of the parameter plane of certain families of rational maps with few critical orbits. Our paradigm is the family R t ( z ) = (1 + (4/27) z 3 /(1 − z)), with dynamics governed by the behaviour of the postcritical orbit ( R n ()) n ∈ℕ . In particular, it is shown that if escapes (that is, R n () tends to infinity), then the Julia set of R is a Cantor set, or a Sierpiński curve, or a curve with one or else infinitely many cut‐points; each of these cases actually occurs.

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