The filled-in Julia set
of a polynomial
is a Julia set and its interior, non-escaping set.
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:
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.