Connectedness locus

In one-dimensional complex dynamics, the connectedness locus is a subset of the parameter space of rational functions, which consists of those parameters for which the corresponding Julia set is connected.

Examples

Without doubt, the most famous connectedness locus is the Mandelbrot set, which arises from the family of complex quadratic polynomials :

The connectedness loci of the higher-degree unicritical families,

(where ) are often called 'Multibrot sets'.

For these families, the bifurcation locus is the boundary of the connectedness locus. This is no longer true in settings, such as the full parameter space of cubic polynomials, where there is more than one free critical point. For these families, even maps with disconnected Julia sets may display nontrivial dynamics. Hence here the connectedness locus is generally of less interest.

gollark: Okay, that is more worrying than anticipated.
gollark: Ah, yes, RSA-3072 and whatever are 3072 *bits*.
gollark: * which fits into 1KiB and for which internal state for bruteforcing fits into the remaining space.
gollark: How would you take over the world with it? You can bruteforce anything which fits into 1KB (are we assuming 1KiB here, not 1000 bytes exactly?), which is not that much.
gollark: "None are safe" is correct as an approximation, not *strictly*.
  • Epstein, Adam; Yampolsky, Michael (March 1999). "Geography of the cubic connectedness locus: Intertwining surgery". Annales Scientifiques de l'École Normale Supérieure. 32 (2): 151–185. arXiv:math/9608213. doi:10.1016/S0012-9593(99)80013-5.
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