Fractals/Iterations in the complex plane/Mandelbrot set/MFpoint12

What is the Myrberg-Feigenbaum point of family ? name MF = the Myrberg-Feigenbaum point the Feigenbaum Pointmuency: feigenbaum pointYouTube: Mikhail Lyubich: Story of the Feigenbaum point. Centre International de Rencontres Mathématiques Accumulation point of period-doubling cascadefractalforums.org: accumulation-point-of-period-doubling images properities The Myrberg-Feigenbaum 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)On Periodic and Chaotic Regions in the Mandelbrot Set by G. Pastor, M. Romera, G. Álvarez, D. Arroyo and F. Montoya the accumulation point is the limit of the disk centers it is a limit of a series of bifurcation parameters ( root points ) of period-2n component. In other words period-doubling cascade finishes at the Myrberg-Feigenbaum point. it is a limit of a series of band-merging points . In other words a period-doubling cascade of chaotic bands also finishes, from the opposite side, at the MF point.EXTERNAL ARGUMENTS FOR THE CHAOTIC BANDS CALCULATION IN THE MANDELBROT SET by G. Pastor , M. Romera, G. Alvarez, and F. Montoya What is the address of Feigenbaum point ? Angled internal address: What is the value of Feigenbaum point ? the Myrberg-Feigenbaum point What external rays land on the Myrberg-Feigenbaum point ? Decimal values of external angles t of rays that lands on the Myrberg-Feigenbaum point are (0.412454... , 0.58755...) How to compute angles of external rays ? To compute angles one can use 2 methods: find a limit of a series of bifurcation parameters ( root points ) of period-2n component. find a limit of a series of band-merging points : How to compute limit of angles landing on bifurcation parameters ? The 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[n-1] 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 ? UNIVERSALIiTY FOR PERIOD n-TUPLINGS IN COMPLEX MAPPINGS by Predrag CVITANOVIC and Jan MYRHEIM. Physics Letters A Volume 94, Issue 8, 28 March 1983, Pages 329-333 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 :Period tripling accumulation point for complexified Henon map by O.B. Isaeva, S.P. Kuznetsov 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 g-gGSK with the complex factor d=4.60022558 -8.98122473i.Effect of noise on the period-tripling by Saratov group of theoretical nonlinear dynamicsScaling properties in dynamics of non-analytic complex maps near the accumulation point of the period-tripling cascade by O.B. Isaeva, S.P. Kuznetsov zoom "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 Decimal expansion of the accumulation point of the logistic map.