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Total disconnectedness of Julia sets of random quadratic polynomials

Part of: Complex dynamical systems Two-dimensional theory

Published online by Cambridge University Press:  04 March 2021

KRZYSZTOF LECH  and
ANNA ZDUNIK
KRZYSZTOF LECH
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097Warsaw, Poland (e-mail: K.Lech@mimuw.edu.pl)
ANNA ZDUNIK*
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097Warsaw, Poland (e-mail: K.Lech@mimuw.edu.pl)
*
e-mail: A.Zdunik@mimuw.edu.pl
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Abstract

For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

Keywords

random dynamicsholomorphic dynamicsJulia setsconnectivity

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 31A05: Harmonic, subharmonic, superharmonic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 5 , May 2022 , pp. 1764 - 1780
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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