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Total disconnectedness of Julia sets of random quadratic polynomials | | Krzysztof Lech
; Anna Zdunik
; | | Date: |
15 Apr 2020 | | Abstract: | For a sequence of complex parameters ${c_n}$ we consider the compositions
of functions $f_{c_n} (z) = z^2 + c_n$, which is 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
Br"uck, B"uger 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 disk. 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 disks, 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. | | Source: | arXiv, 2004.6955 | | Services: | Forum | Review | PDF | Favorites | |
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