Julia set

E705565

fractal mathematical object subset of the complex plane

A Julia set is a complex fractal formed by iterating a function on the complex plane, often producing intricate, self-similar boundary patterns that are central objects in complex dynamics.

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Observed surface forms (1)

Surface form	Occurrences
Julia sets	1

Statements (50)

Predicate	Object
instanceOf	fractal ⓘ mathematical object ⓘ subset of the complex plane ⓘ
definedOn	complex plane ⓘ
dimension	Hausdorff dimension often strictly greater than topological dimension ⓘ
field	complex analysis ⓘ complex dynamics ⓘ dynamical systems ⓘ
hasProperty	boundary of Fatou set ⓘ can be a Cantor set ⓘ can be connected or disconnected ⓘ can have non-integer Hausdorff dimension ⓘ can have positive area for some maps ⓘ closed set ⓘ dense periodic points ⓘ nowhere dense in many cases ⓘ often exhibits chaotic dynamics ⓘ often fractal boundary ⓘ perfect set ⓘ self-similar ⓘ sensitive dependence on initial conditions ⓘ topologically transitive dynamics ⓘ totally invariant under the function ⓘ
historicalDevelopment	further developed by Pierre Fatou ⓘ studied by Gaston Julia in early 20th century ⓘ
invarianceProperty	backward invariant under the defining map ⓘ completely invariant under the defining map ⓘ forward invariant under the defining map ⓘ
isDefinedAs	closure of repelling periodic points of a rational map ⓘ complement of the Fatou set ⓘ set of points with chaotic behavior under iteration of a complex function ⓘ
isDefinedFor	polynomials in one complex variable ⓘ rational maps on the Riemann sphere ⓘ transcendental entire functions ⓘ
namedAfter	Gaston Julia NERFINISHED ⓘ
relatedTo	Fatou set NERFINISHED ⓘ Mandelbrot set NERFINISHED ⓘ iteration of complex functions ⓘ polynomial map ⓘ rational map ⓘ
topology	can be a dendrite (locally connected continuum without simple closed curves) ⓘ can be a simple closed curve for some polynomials ⓘ can be totally disconnected ⓘ
typicalExample	Julia set of z^2 + c NERFINISHED ⓘ quadratic Julia set ⓘ
usedIn	mathematical art ⓘ study of chaotic dynamical systems ⓘ visualization of fractal geometry ⓘ
visualization	often colored by iteration count before escape ⓘ often rendered by escape-time algorithm ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Lyapunov fractal → relatedTo → Julia set ⓘ

Dynamics in One Complex Variable → topic → Julia set ⓘ

this entity surface form: Julia sets