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A New Simple Class of Rational Functions whose Julia Set is the whole Riemann Sphere
Author(s) -
Inninger Clemens,
Peherstorfer Franz
Publication year - 2002
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/s0024610701003027
Subject(s) - julia set , riemann sphere , mathematics , newton fractal , rational function , schwarzian derivative , pure mathematics , simple (philosophy) , set (abstract data type) , meromorphic function , function (biology) , class (philosophy) , riemann surface , mathematical analysis , computer science , mathematical optimization , local convergence , philosophy , iterative method , epistemology , evolutionary biology , biology , programming language , artificial intelligence
The paper first gives sufficient conditions on the critical points and the Schwarzian derivative of a real rational function R such that the Julia set of R is C ¯ . Further, it is shown that under mild conditions on another real rational function R ˜ with possibly non‐empty Fatou set, the Julia set of R ˜ ∘ R is the whole Riemann sphere again. Then families of rational functions are given whose Julia set is C ¯ and whose critical points are not necessarily preperiodic. Concrete examples were previously available only for the preperiodic case. Finally, it is demonstrated that the methods presented also apply to the construction of polynomials whose Julia sets are dendrites and whose critical points in the Julia set are not necessarily preperiodic.