Fatou sets
E911368
Fatou sets are the regions in the complex plane where the iterates of a complex function behave in a stable, regular manner, forming the complement of the chaotic Julia set in complex dynamics.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Fatou set | 0 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
object in complex dynamics ⓘ |
| behaviorOfIterates |
dynamics is stable under small perturbations of initial point
ⓘ
iterates form a normal family ⓘ orbits depend continuously on initial conditions ⓘ |
| characterizedBy |
equicontinuity of iterates on compact subsets
ⓘ
normality of the family of iterates ⓘ stable behavior of iterates ⓘ |
| complementOf | Julia set NERFINISHED ⓘ |
| context |
holomorphic iteration on the Riemann sphere
ⓘ
holomorphic iteration on the complex plane ⓘ |
| contrastedWith | Julia set ⓘ |
| definedFor |
iterated holomorphic function
ⓘ
meromorphic function ⓘ rational map on the Riemann sphere ⓘ |
| dependsOn |
dynamical system on the complex plane
ⓘ
iterated function ⓘ |
| exampleFor |
exponential map e^z
ⓘ
quadratic polynomial z^2 + c ⓘ |
| field |
complex analysis
ⓘ
complex dynamics ⓘ |
| hasBoundary | Julia set in many classical cases ⓘ |
| hasProperty |
backward invariant under the function
ⓘ
can have infinitely many components ⓘ components are maximal domains of normality ⓘ forward invariant under the function ⓘ may be empty for some maps ⓘ open set ⓘ union of Fatou components ⓘ |
| introducedBy | Pierre Fatou NERFINISHED ⓘ |
| isSubsetOf |
Riemann sphere
NERFINISHED
ⓘ
complex plane ⓘ |
| namedAfter | Pierre Fatou NERFINISHED ⓘ |
| relatedConcept |
Fatou component
NERFINISHED
ⓘ
Julia set ⓘ Montel theorem NERFINISHED ⓘ normal family ⓘ |
| studiedIn |
iteration theory of entire functions
ⓘ
iteration theory of rational maps ⓘ |
| topologicalDecomposition | connected components called Fatou components ⓘ |
| typicalDynamics |
Baker domains
ⓘ
Herman rings ⓘ Siegel disks ⓘ attracting cycles ⓘ parabolic cycles ⓘ wandering domains ⓘ |
| usedIn | classification of dynamical behavior of complex maps ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.