Fractals/Iterations in the complex plane/Fatou set
Definition
[edit | edit source]The Fatou set is called:
- the domain of normality
- the domain of equicontinuity
Parts
[edit | edit source]Fatou set, domains and components Then there is a finite number of open sets , that are left invariant by and are such that:
- the union of the sets is dense in the plane and
- behaves in a regular and equal way on each of the sets
The last statement means that the termini of the sequences of iterations generated by the points of are
- either precisely the same set, which is then a finite cycle
- or they are finite cycles of circular or annular shaped sets that are lying concentrically.
In the first case the cycle is attracting, in the second it is neutral.
These sets are the Fatou domains of and their union is the Fatou set of
Each domain of the Fatou set of a rational map can be classified into one of four different classes.[1]
Each of the Fatou domains contains at least one critical point of that is:
- a (finite) point z satisfying
-
- if the degree of the numerator is at least two larger than the degree of the denominator
- if for some c and a rational function satisfying this condition.
Complement
[edit | edit source]The complement of is the Julia set of
If all the critical points are preperiodic, that is they are not periodic but eventually land on a periodic cycle, then is all the sphere; otherwise, is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). Like is left invariant by and on this set the iteration is repelling, meaning that for all w in a neighbourhood of z [within ]. This means that behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and they make up an infinitesimal part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.
components
[edit | edit source]Number of Fatou set's components in case of rational map:[2]
- 0 ( Fatou set is empty, the whole Riemann sphere is a Julia set )[3]
- 1 ( example , here is only one Fatou domains which consist of one component = full Fatou set)
- 2 ( example , here are 2 Fatou domains, both have one component )
- infinitely many ( example , here are 2 Fatou domains, one ( the exterior) has one component, the other ( interior) has infinitely many components)
-
2 domains ( basins) and 2 components, so domain = component
-
2 domains: 1 domain has one component, but the other domain has infinitely many components
-
1 domain and infinitely many components
the Julia set for the The Samuel Lattes function consists of the whole complex sphere = Fatou set is empty[4][5]
domains
[edit | edit source]In case of discrete dynamical system based on complex quadratic polynomial Fatou set can consist of domains ( basins) :
- attracting ( basin of attraction of fixed point / cycle )
- superattracting ( Boettcher coordinate )
- basin of infinity [6]
- attracting but not superattracting (
- superattracting ( Boettcher coordinate )
- parabolic (Leau-Fatou) basin ( Fatou coordinate ) Local dynamics near rationally indifferent fixed point/cycle ;
- elliptic basin = Siegel disc ( Local dynamics near irrationally indifferent fixed point/cycle )
coordinate
[edit | edit source]Basin | coordinate | speed of approach to fixed point | |
---|---|---|---|
superattracting | Boettcher | fast | |
attracting | Koenigs | intermediate | |
parabolic | Fatou | slow | |
Siegel disc | irrational rotation so orbits are closed curves | point rotates around fixed point and never approaches it | |
Herman ring | irrational rotation so orbits are closed curves | point rotates around fixed point and never approaches it |
Repelling basin is another name for
- superattracting basin for polynomials
Local discrete complex dynamics
[edit | edit source]Julia set is connected ( 2 basins of attraction)
- attracting : hyperbolic dynamics
- superattracting : the very fast ( = exponential) convergence to periodic cycle ( fixed point )
- parabolic component = slow ( lazy ) dynamics = slow ( exponential slowdown) convergence to parabolic fixed point ( periodic cycle)
- Siegel disc component = rotation around fixed point and never reach the fixed point
When Julia set is disconnected ther is no interior of Julia set ( critical fixed point is repelling ( or attracting to infinity) - onlu one basin of attraction
parameter c | location of c | Julia set | interior | type of critical orbit dynamics | critical point | fixed points | stability of alfa |
---|---|---|---|---|---|---|---|
c = 0 | center, interior | connected | exist | superattracting | atracted to alfa fixed point | fixed critical point equal to alfa fixed point, alfa is superattracting, beta is repelling | r = 0 |
0<c<1/4 | internal ray 0, interior | connected | exist | attracting | atracted to alfa fixed point | alfa is attracting, beta is repelling | 0 < r < 1.0 |
c = 1/4 | cusp, boundary | connected | exist | parabolic | atracted to alfa fixed point | alfa fixed point equal to beta fixed point, both are parabolic | r = 1 |
c>1/4 | external ray 0, exterior | disconnected | disappears | repelling | repelling to infinity | both finite fixed points are repelling | r > 1 |
Stability r is absolute value of multiplier at fixed point alfa:
c = 0.0000000000000000+0.0000000000000000*I m(c) = 0.0000000000000000+0.0000000000000000*I r(m) = 0.0000000000000000 t(m) = 0.0000000000000000 period = 1 c = 0.0250000000000000+0.0000000000000000*I m(c) = 0.0513167019494862+0.0000000000000000*I r(m) = 0.0513167019494862 t(m) = 0.0000000000000000 period = 1 c = 0.0500000000000000+0.0000000000000000*I m(c) = 0.1055728090000841+0.0000000000000000*I r(m) = 0.1055728090000841 t(m) = 0.0000000000000000 period = 1 c = 0.0750000000000000+0.0000000000000000*I m(c) = 0.1633399734659244+0.0000000000000000*I r(m) = 0.1633399734659244 t(m) = 0.0000000000000000 period = 1 c = 0.1000000000000000+0.0000000000000000*I m(c) = 0.2254033307585166+0.0000000000000000*I r(m) = 0.2254033307585166 t(m) = 0.0000000000000000 period = 1 c = 0.1250000000000000+0.0000000000000000*I m(c) = 0.2928932188134524+0.0000000000000000*I r(m) = 0.2928932188134524 t(m) = 0.0000000000000000 period = 1 c = 0.1500000000000000+0.0000000000000000*I m(c) = 0.3675444679663241+0.0000000000000000*I r(m) = 0.3675444679663241 t(m) = 0.0000000000000000 period = 1 c = 0.1750000000000000+0.0000000000000000*I m(c) = 0.4522774424948338+0.0000000000000000*I r(m) = 0.4522774424948338 t(m) = 0.0000000000000000 period = 1 c = 0.2000000000000000+0.0000000000000000*I m(c) = 0.5527864045000419+0.0000000000000000*I r(m) = 0.5527864045000419 t(m) = 0.0000000000000000 period = 1 c = 0.2250000000000000+0.0000000000000000*I m(c) = 0.6837722339831620+0.0000000000000000*I r(m) = 0.6837722339831620 t(m) = 0.0000000000000000 period = 1 c = 0.2500000000000000+0.0000000000000000*I m(c) = 0.9999999894632878+0.0000000000000000*I r(m) = 0.9999999894632878 t(m) = 0.0000000000000000 period = 1 c = 0.2750000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.3162277660168377*I r(m) = 1.0488088481701514 t(m) = 0.0487455572605341 period = 1 c = 0.3000000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.4472135954999579*I r(m) = 1.0954451150103321 t(m) = 0.0669301182003075 period = 1 c = 0.3250000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.5477225575051662*I r(m) = 1.1401754250991381 t(m) = 0.0797514300099943 period = 1 c = 0.3500000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.6324555320336760*I r(m) = 1.1832159566199232 t(m) = 0.0897542589928440 period = 1
-
center = superattracting
-
attracting
-
parabolic
-
repelling
Tests
[edit | edit source]- drawing critical orbit(s)
- finding periodic points
- dividing complex move into simple paths
- topological graph,[7]
- drawing grid ( polar or rectangular )
method | test | description | resulting sets | true sets | |
---|---|---|---|---|---|
binary escape time | bailout | abs(zn)>ER | escaping and not escaping | Escaping set contains fast escaping pixels and is a true exterior.
Not escaping set is treated as a filled Julia set ( interior and boundary) but it contains :
| |
discrete escape time = Level Set Method = LSM | bailout | Last iteration or final_n = n : abs(zn)>ER | escaping set is divided into subsets with the same n ( last iteration). This subsets are called Level Sets and create bands surrounding and approximating Julia set. Boundaries of level sets are called dwell-bands | ||
continous escape time | Example | Example | Example |
Tests
- for finite attractors ( radius = AR)
- for ininity ( bailout test with Escaping Radiud ( ER)
Targets
[edit | edit source]- trap for forward orbit
- it is a set which captures any orbit tending to fixed / periodic point\
- is always inside component containing fixed point
-
-
parabolic case
-
parabolic case
References
[edit | edit source]- ↑ Beardon, Iteration of Rational Functions, Theorem 7.1.1.
- ↑ Beardon, Iteration of Rational Functions, Theorem 5.6.2.
- ↑ Campbell, J.T., Collins, J.T. Blowup Points and Baby Mandelbrot Sets for a Family of Singularly Perturbed Rational Maps. Qual. Theory Dyn. Syst. 16, 31–52 (2017). https://doi.org/10.1007/s12346-015-0169-5
- ↑ Maru Sarazola : On the cardinalities of the Fatou and Julia sets
- ↑ Complex Dynamics by Fionn´an Howard
- ↑ THE CLASSIFICATION OF POLYNOMIAL BASINS OF INFINITY by LAURA DEMARCO AND KEVIN PILGRIM
- ↑ A Topology Simplification Method For 2D Vector Fields by Xavier Tricoche, Gerik Scheuermann and Hans Hagen