Article contents
Total disconnectedness of Julia sets of random quadratic polynomials
Published online by Cambridge University Press: 04 March 2021
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.
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- Original Article
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- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
References
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