The filled-in Julia set
of a polynomial
is a Julia set and its interior, non-escaping set.
Formal definition
The filled-in Julia set
of a polynomial
is defined as the set of all points
of the dynamical plane that have bounded orbit with respect to
![{\displaystyle K(f){\overset {\mathrm {def} }{{}={}}}\left\{z\in \mathbb {C} :f^{(k)}(z)\not \to \infty ~{\text{as}}~k\to \infty \right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35e83a188a919e7926fdba5eb4a67f634f0c4bc1)
where:
Relation to the Fatou set
The filled-in Julia set is the (absolute) complement of the attractive basin of infinity.
![{\displaystyle K(f)=\mathbb {C} \setminus A_{f}(\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2a55f10534f48b61c1d5562f36a06ec01e26e4)
The attractive basin of infinity is one of the components of the Fatou set.
![{\displaystyle A_{f}(\infty )=F_{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efcf935bbc7808b2adf46b4752bba79362ecc5fd)
In other words, the filled-in Julia set is the complement of the unbounded Fatou component:
![{\displaystyle K(f)=F_{\infty }^{C}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6612c7bb195a239a46645198a9bafd1f3f07071)
Relation between Julia, filled-in Julia set and attractive basin of infinity
Wikibooks has a book on the topic of:
Fractals
The Julia set is the common boundary of the filled-in Julia set and the attractive basin of infinity
![{\displaystyle J(f)=\partial K(f)=\partial A_{f}(\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4f8e331e1cdf635499b7619afd74cdae9857e2)
where:
![{\displaystyle A_{f}(\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a59ad0e68bfaf7a758fdb65c90a777324d9dd526)
denotes the
attractive basin of
infinity = exterior of filled-in Julia set = set of escaping points for
![{\displaystyle A_{f}(\infty )\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\to \infty \ as\ k\to \infty \}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d92000e8bba277f3ff047f79177e5756728d1ded)
If the filled-in Julia set has no interior then the Julia set coincides with the filled-in Julia set. This happens when all the critical points of
are pre-periodic. Such critical points are often called Misiurewicz points.
Spine
The most studied polynomials are probably those of the form
, which are often denoted by
, where
is any complex number. In this case, the spine
of the filled Julia set
is defined as arc between
-fixed point and
,
![{\displaystyle S_{c}=\left[-\beta ,\beta \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc28d767679ab7c75c23b245f5213dc2e38cb52)
with such properties:
- spine lies inside
.[1] This makes sense when
is connected and full[2]
- spine is invariant under 180 degree rotation,
- spine is a finite topological tree,
- Critical point
always belongs to the spine.[3]
-fixed point is a landing point of external ray of angle zero
,
is landing point of external ray
.
Algorithms for constructing the spine:
- detailed version is described by A. Douady[4]
- Simplified version of algorithm:
- connect
and
within
by an arc,
- when
has empty interior then arc is unique,
- otherwise take the shortest way that contains
.[5]
Curve
:
![{\displaystyle R{\overset {\mathrm {def} }{{}={}}}R_{1/2}\cup S_{c}\cup R_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae47b5b210fc4afd7a91e2526f551021f2c47e9)
divides dynamical plane into two components.
Images
-
Filled Julia set for f
c, c=1−φ=−0.618033988749…, where φ is the
Golden ratio
-
Filled Julia with no interior = Julia set. It is for c=i.
-
Filled Julia set for c=−1+0.1*i. Here Julia set is the boundary of filled-in Julia set.
-
-
Filled Julia set for c = −0.8 + 0.156i.
-
Filled Julia set for c = 0.285 + 0.01i.
-
Filled Julia set for c = −1.476.
Names
Notes
- ^ Douglas C. Ravenel : External angles in the Mandelbrot set: the work of Douady and Hubbard. University of Rochester Archived 2012-02-08 at the Wayback Machine
- ^ John Milnor : Pasting Together Julia Sets: A Worked Out Example of Mating. Experimental Mathematics Volume 13 (2004)
- ^ Saaed Zakeri: Biaccessiblility in quadratic Julia sets I: The locally-connected case
- ^ A. Douady, “Algorithms for computing angles in the Mandelbrot set,” in Chaotic Dynamics and Fractals, M. Barnsley and S. G. Demko, Eds., vol. 2 of Notes and Reports in Mathematics in Science and Engineering, pp. 155–168, Academic Press, Atlanta, Georgia, USA, 1986.
- ^ K M. Brucks, H Bruin : Topics from One-Dimensional Dynamics Series: London Mathematical Society Student Texts (No. 62) page 257
- ^ The Mandelbrot Set And Its Associated Julia Sets by Hermann Karcher
References
- Peitgen Heinz-Otto, Richter, P.H. : The beauty of fractals: Images of Complex Dynamical Systems. Springer-Verlag 1986. ISBN 978-0-387-15851-8.
- Bodil Branner : Holomorphic dynamical systems in the complex plane. Department of Mathematics Technical University of Denmark, MAT-Report no. 1996-42.