Julia set
Let be an open subset of the complex plane and let be analytic. Denote the -th iterate of by , i.e. and . Then the Julia set of is the subset of characterized by the following property: if then the restriction of to any neighborhood of is not a normal family.
It can also be shown that the Julia set of is the closure of the set of repelling periodic points of . (Repelling periodic point means that, for some , we have and .)
A simple example is afforded by the map ; in this case, the Julia set is the unit circle. In general, however, things are much more complicated and the Julia set is a fractal.
From the definition, it follows that the Julia set is closed under and its inverse — and . Topologically, Julia sets are perfect and have empty interior.
Title | Julia set |
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Canonical name | JuliaSet |
Date of creation | 2013-03-22 17:15:26 |
Last modified on | 2013-03-22 17:15:26 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 7 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 28A80 |