Fractals/Iterations in the complex plane/Mandelbrot set/MFpoint12
What is the MyrbergFeigenbaum point of family ?
name
edit MF = the MyrbergFeigenbaum point
 the Feigenbaum Point^{[1]}^{[2]}
 Accumulation point of perioddoubling cascade^{[3]}
images
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Zoom toward Feigenbaum point

The Feigenbaum point (red arrow) is limit of the bifurcation point

Feigenbaum point Julia set
properities
editThe MyrbergFeigenbaum point is
 a point c of parameter plane
 a Misiurewicz point
 a biaccesible point. It means that it is a landing point of 2 external rays with irrational angles. The rays are not spiralling at all (no turn), because if the Misiurewicz point is a real number, it does not turn at all
 boundary point between chaotic (2 < c < MF) and periodic region (MF< c < 1/4)^{[4]}
 the accumulation point is the limit of the disk centers
 it is a limit of a series of bifurcation parameters ( root points ) of period2^{n} component. In other words perioddoubling cascade finishes at the MyrbergFeigenbaum point.
 it is a limit of a series of bandmerging points . In other words a perioddoubling cascade of chaotic bands also finishes, from the opposite side, at the MF point.^{[5]}
What is the address of Feigenbaum point ?
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What is the value of Feigenbaum point ?
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What external rays land on the MyrbergFeigenbaum point ?
editDecimal values of external angles t of rays that lands on the MyrbergFeigenbaum point are (0.412454... , 0.58755...)
How to compute angles of external rays ?
editTo compute angles one can use 2 methods:
 find a limit of a series of bifurcation parameters ( root points ) of period2^{n} component.
 find a limit of a series of bandmerging points :
How to compute limit of angles landing on bifurcation parameters ?
editThe candidate upper external angle is obtained by using the substitution (string replacing): 0 > 01 and 1 > 10 repeatedly:
 0
 01
 0110
 01101001
 0110100110010110
 ...
But it is not known whether the rays actually lands; maybe M is not locally connected at the Feigenbaum point and some long decorations are shielding it from external rays.
One can compute it using Maxima CAS program :
kill(all); remvalue(all); f(x):=if (x=0) then [0,1] else [1,0]; compile(all); a:[]; a:endcons([0],a); for n:2 thru 10 step 1 do ( a:endcons([],a), for x in a[n1] do ( a[n]:endcons(first(f(x)),a[n]), a[n]:endcons(second(f(x)),a[n])), print(n,a[n]) );
How to compute points of perid n tupling bifurcations ?
edit UNIVERSALIiTY FOR PERIOD nTUPLINGS IN COMPLEX MAPPINGS by Predrag CVITANOVIC and Jan MYRHEIM. Physics Letters A Volume 94, Issue 8, 28 March 1983, Pages 329333
 period tripling:
 point of Golberg  Sinai  Khanin (GSK) lGSK = 0.0236411685 + 0.7836606508i
 Corresponded critical point (GSK point) is situated at λc = 0.0236411685377 + 0.7836606508052i and characterized by following critical indexes [2], namely, by critical multiplier µc, scale factor α, and parameter scaling constant δ :^{[6]}
 µc = −0.47653179 − 1.05480867i
 α = −2.0969 + 2.3583i
 δ = 4.6002 − 8.9812i
 Near the critical point GSK a structure of "leaves" of the Mandelbrot set possesses a property of scale invariance in respect to rescaling of ggGSK with the complex factor d=4.60022558 8.98122473i.^{[7]}^{[8]}
zoom
edit "sequence of illustrations, each view is centered at the Feigenbaum point and the magnification increases by 4.6692 (the Feigenbaum Constant) each time. The filaments become steadily denser until they fill the view."
References
edit ↑ muency: feigenbaum point
 ↑ YouTube: Mikhail Lyubich: Story of the Feigenbaum point. Centre International de Rencontres Mathématiques
 ↑ fractalforums.org: accumulationpointofperioddoubling
 ↑ On Periodic and Chaotic Regions in the Mandelbrot Set by G. Pastor, M. Romera, G. Álvarez, D. Arroyo and F. Montoya
 ↑ EXTERNAL ARGUMENTS FOR THE CHAOTIC BANDS CALCULATION IN THE MANDELBROT SET by G. Pastor , M. Romera, G. Alvarez, and F. Montoya
 ↑ Period tripling accumulation point for complexified Henon map by O.B. Isaeva, S.P. Kuznetsov
 ↑ Effect of noise on the periodtripling by Saratov group of theoretical nonlinear dynamics
 ↑ Scaling properties in dynamics of nonanalytic complex maps near the accumulation point of the periodtripling cascade by O.B. Isaeva, S.P. Kuznetsov