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 eta as branchpoint of tetrational mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 06/10/2011, 09:36 PM (06/10/2011, 01:54 PM)sheldonison Wrote: Looks great! Thanks for making the graph. I'm not sure I understand all of your comments though... regular tetration at a base>eta would be a complex superfunction, with no real valued real axis, right? Regular iteration is complex for complex bases, but I'm talking about the bipolar/perturbed-Fatou/Kneser iteration. If this "merge" tetrational can be considered as the analytic continuation of the Kneser tetrational halfway around the branch point at $b = \eta$ (the two functions graphed, then, are, respectively, the continuation around the branch point through the lower and upper half-plane to the real axis where $b < \eta$, specifically $b = \sqrt{2}$), then since it is sooooooo close to being real-valued for real heights, the branch point at $b = \eta$ must be extremely slight, at least for real heights greater than -2 and when the cut is placed at the real axis for $b < \eta$. It is like how there is a "mild" singularity/branch point in $\exp^{1/2}(z)$ at the conjugate fixed point pair, but perhaps even "milder". sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 06/10/2011, 11:00 PM (06/10/2011, 09:36 PM)mike3 Wrote: Regular iteration is complex for complex bases, but I'm talking about the bipolar/perturbed-Fatou/Kneser iteration. If this "merge" tetrational can be considered as the analytic continuation of the Kneser tetrational halfway around the branch point at $b = \eta$ (the two functions graphed, then, are, respectively, the continuation around the branch point through the lower and upper half-plane to the real axis where $b < \eta$, specifically $b = \sqrt{2}$), then since it is sooooooo close to being real-valued for real heights, the branch point at $b = \eta$ must be extremely slight, at least for real heights greater than -2 and when the cut is placed at the real axis for $b < \eta$. It is like how there is a "mild" singularity/branch point in $\exp^{1/2}(z)$ at the conjugate fixed point pair, but perhaps even "milder".Nicely stated. I don't think we know what happens on the Shell Thron region itself, except at base eta. For bases<>eta, on the boundary, we have a repelling fixed point, and a parabolic fixed point. - Sheldon sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 06/13/2011, 05:14 AM (This post was last modified: 06/13/2011, 05:54 AM by sheldonison.) (06/10/2011, 11:00 PM)sheldonison Wrote: Nicely stated. I don't think we know what happens on the Shell Thron region itself, except at base eta. For bases<>eta, on the boundary, we have a repelling fixed point, and a parabolic fixed point. - SheldonThis is a repost. The question is, at the Shel-Thron Boundary, can we develop a merged Fatou like mapping from the two fixed points? Originally, I was going to post that there are two fixed points on the Shel Thron boundary, but that only one of these could be used to develop an analytic superfunction; except now I'm pretty sure that both can be used to develop an analytic superfunction. Imagine following a circular arc around $\eta$ starting from base $\sqrt{2}$ and continuing along to base $2\eta-\sqrt{2}$. Then along this arc, the repelling fixed point which starts at L2=4 morphs seamlessly, and remains a repelling fixed point for the entire path, and becomes the complex conjugate fixed point, with Im(L2)<0. The superfunction from L2 can be calculated anywhere on this path, with known methods. The attracting fixed point starts out at L2=2. But when you cross the Shell Thron region (at the base given below), the attracting fixed point becomes neither attracting, nor repelling, nor parabolic. It becomes a neutral circular fixed point, with a real Period. Because the absolute value of the log of L1=1, this fixed point is not repelling, and not attracting, but rather circular. From the wiki page, "The boundary of the Shell-Thron region ... where |log(L1)|=1". Code:Results at the Shel Thron boundary base  = 1.4749205843145635852862061112 + 0.0034981478091858060388588917267i   L1 =  (circular)   2.4372341100745143830388282062 + 0.82552153847929520132720783617i L2 =  (repelling)  2.5206834347439820199663539286 - 0.87677210299493726158947155475i |log(L1)|=1.0000000000000000000000000000 (circular) |log(L2)|=1.0371394570002141836820569537 (repelling) Period1=  18.886173137606161572331556476 (real period) k      =  0.33268705424861870835170403130i Period2= -18.886314043722660117719560870 + 2.0956661382058037697500756852iNow, assume that f is the superfunction from the L1 circular fixed point. If you take a point very close to the L1 fixed point, and use that sequence of points, all very close to the L1 fixed point, to try to develop a Superfunction, then you will find that the sequence of points has a real period, which I originally thought was nonsense, but now I think it is probably possible to develop an analytic superfunction at the L1 fixed point, although I don't know how to calculate it yet. $f(z)=L1+\delta$ $f(z+1)=B^{f(z)}=B^{L1+\delta}$ $B^{L1+\delta}\approx L1+\delta\log(L1) = L1+\delta\exp(\log(\log(L1))) = L1+\delta \exp(k)$ where $k=\log(\log(L1))$, where $\Re(k)=0$ (as shown below) $f(z+2) \approx L1+\delta\exp(2k)$ $f(z+3) \approx L1+\delta\exp(3k)$.... It turns out that k is an imaginary number, $k=\log(\log(L1))$, where by definition $|\log(L1)|$=1. K is linked to the Period by the equation $k=\log(\log(L1))=2\pi i/\text{Period1}$. Note that $\Re(k)=\Re(\log(\log(L1)))=0$, since |log(L1)|=1. So then k is an imaginary number, and Period1 is a real number. $f(z) \approx L1+\exp(zk)$, where exp(zk) decays to zero as $\Im(z)$ increases. My conjecture is that this might work with an infinite sequence of terms, all decaying as $\Im(z)$ increases. $\text{SuperFunction}_{L1}(z) = \sum_{n=1}^{\infty} a_n\exp(2n\pi iz/\text{Period1})$ So then we would have $f(z+\text{Period1}) \approx f(z+\exp(2n\pi i z/ \text{Period1})) \approx f(z+\exp(2\pi i)) \approx f(z)$, which is an exact real periodic equation, if an infinite number of terms is used for the superfunction. When I used only one term for the real periodic superfunction, the error term had twice the frequency, and was magnitude squared (smaller, as imag(z) increased). So, the question is, how to calculate the infinite sequence of terms, and does this work? At the Shel-Thron boundary, the other fixed point has $\Im(L2)<0$. Is it possible to somehow merge the two fixed point functions with a Fatou merger, or with a Kneser mapping? - Sheldon sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 06/14/2011, 03:19 PM (This post was last modified: 06/20/2011, 09:32 PM by sheldonison.) Continuing the discussion from my previous post, where we consider the behavior of the superfunction along a circular arc, from base=$\sqrt{2}$ to base=$2\eta-\sqrt{2}$. For a complex base at the Shell-Thron Boundary, how does the superfunction evolve as we circle closely around base=$\eta$? And also, can we develop a merged Fatou like mapping from the two fixed points at that base? Code:results for a complex base at the Shell-Thron boundary base  = 1.4749205843145635852862061112 + 0.0034981478091858060388588917267i   L1    = 2.4372341100745143830388282062 + 0.82552153847929520132720783617i (circular fixed point) |log(L1)|=1.0000000000000000000000000000 (circular) Period1=  18.886173137606161572331556476 (real period) L2    = 2.5206834347439820199663539286 - 0.87677210299493726158947155475i (repelling fixed point at this base)This L1 fixed point superfunction at the Shell-Thron boundary, with a real period=18.89 is very interesting. My calculations match the mathematical model of the superfunction from L1 being a periodic function, with a real period of approximately 18.89, which decays to L1 as $z\to+i\infty$. I generated the Fourier series for the function, and I also generated a plot of the sexp(z) function for this base. To generate the sexp(z) plot, I started with z=1, and iterating z=B^z, for 16000 iterations, subtracting iterations of the real Period. The pattern repeats with a real period of 18.89 to +/- real infinity. Notice that sexp(0)=1, sexp(-1)=0, and notice the approximation of the singularity at z=-2, where we would have log(0). My assumption is that none of the derivatives of the function here are defined anywhere in this cross section, but that for imag(z)>singularity(z), the function is analytic. As the radius approaches closer and closer to eta, this function's real period goes to infinity, and this function becomes the lower superfunction for base eta.     $\text{SuperFunction}_{L1}(z) = L1 + \sum_{n=1}^{\infty} a_n\exp(2n\pi iz/\text{Period1})$ Here are the first 80 theta(z) Fourier series terms, which is accurate to 32 decimal digits for imag(z)>=1.5i. This Fourier series for the superfunction, was developed with the somewhat arbitrary requirement that the a_1 term=1. As $z\to i\infty$, the superfunction decays to a periodic function with a period of ~18.89, circling L1 with an exponentially shrinking radius. At imag(z)=-i, which appears to be just slightly above the sexp(z) fractal singularity plot, the superfunction approximation is still reasonably accurate, to +/- 0.1 for most points, and approximates the sexp(z) plot above. Code:a1=   1 a2=  -0.0936201434689961368086647459633896 - 0.579238923230953573245678436083305*I a3=  -0.320683212371393341813787362082673 + 0.126743588529914531522707124081828*I a4=   0.119616793864212915772264614599621 + 0.164007639553429767557332132175408*I a5=   0.0732389126380630280505692013728790 - 0.0946974356166440404102912124319164*I a6=  -0.0663742380831858892066723849627545 - 0.0244308418378105429681555493459044*I a7=  -0.00113917177691053516060857128440486 + 0.0418461132213518228505660201678252*I a8=   0.0235941882293188633555260249973232 - 0.00770820240348467892917041034129283*I a9=  -0.00920933800613021623902496158444828 - 0.0115141419178171674893277574536941*I a10= -0.00437869312492190154118567427835918 + 0.00760287408841178916234910646887856*I a11=  0.00518243461543642777367630749190422 + 0.000703172062265366709132895289567522*I a12= -0.000822335040668600992619235143434256 - 0.00301247557659202033418543708431202*I a13= -0.00145021915279076526248432300509339 + 0.00117634654885793149100658597647531*I a14=  0.00100173652080411074133996349716316 + 0.000496505353409563045900142392722522*I a15=  0.0000106106561384327391968341507990486 - 0.000669811132053360353384377725305466*I a16= -0.000362051071335304189862533507630806 + 0.000173006216111239867118005085654564*I a17=  0.000191999420945188253150066666075303 + 0.000143514630594805499331330577520006*I a18=  0.0000180105057854429502171089437309786 - 0.000140692046300459536323603068136406*I a19= -0.0000722817144413949442010819146488065 + 0.0000353580254861441148050161200387193*I a20=  0.0000623843250976460079037027858671792 + 0.0000622994206876068031507183802208773*I a21=  0.0000394144279898223425421990281491789 - 0.0000644399032387350978471352209618663*I a22= -0.0000552399546784824284020360655198103 - 0.0000173717300761573123065860634846381*I a23= -0.00000170900299301120308864944637450145 + 0.0000417901226616108337504224748852074*I a24=  0.0000282825602470756133396057950556508 - 0.00000700465114159728374550033785950976*I a25= -0.0000102741251642115188421455104766420 - 0.0000169513861018562623392242118417184*I a26= -0.00000861450276185261307846582779272648 + 0.0000100872080731087760276251983926557*I a27=  0.00000817321818082630500200232724077194 + 0.00000318851816155030353724964963717565*I a28=  0.000000128226838107579182941357184499617 - 0.00000575088366455444677949785380908752*I a29= -0.00000353092927538484212483341617268819 + 0.00000125499228716754783926743818230882*I a30=  0.00000160003688794248510453112332724949 + 0.00000182794597930009973297955250749624*I a31=  0.000000697557688269020001469882035409570 - 0.00000140542576055879895912891047819070*I a32= -0.00000100889900661331713626426196439158 - 0.0000000564465426616676164278197787755211*I a33=  0.000000230704122240189039616621944108059 + 0.000000606898414029889666096531558352536*I a34=  0.000000290650131894774291241762981855498 - 0.000000298068834574435782124277390943904*I a35= -0.000000252796710132249881563394657044474 - 0.0000000838064588282311556318206599351208*I a36=  0.0000000262525023567587387196047136774841 + 0.000000167679864301056914917435645077030*I a37=  0.0000000843156787032266336785583097242438 - 0.0000000661239762044488524011014511173683*I a38= -0.0000000617506417564549303284536829588438 - 0.0000000200216896839990662943508218014508*I a39=  0.00000000435332496521962851123056565020908 + 0.0000000571323648825386598048750467464987*I a40=  0.0000000429334793323671365061159929480907 - 0.0000000166404992504174499695999020022156*I a41= -0.0000000209040338191777245365860609109214 - 0.0000000275465438797457343240840060614878*I a42= -0.0000000148019299709549500031744130735256 + 0.0000000200070426208339939250862704370370*I a43=  0.0000000164127232147400360617236888494594 + 0.00000000583426861205331198249205244563447*I a44=  0.000000000405211095766879927886426992315563 - 0.0000000119547707093560428329006970617555*I a45= -0.00000000777496117032379804962737474521166 + 0.00000000228864503143700362724336343241042*I a46=  0.00000000315506457738290530801554495546432 + 0.00000000443047184583984061051276451431495*I a47=  0.00000000206832355615669591012934756235558 - 0.00000000297689533811372157097801754876968*I a48= -0.00000000232884755383860191534522210742407 - 0.000000000595568786314963087888383006153234*I a49=  1.87943022217748645613168888408468 E-10 + 0.00000000157724751299808970393836445878456*I a50=  0.000000000918930069471866341849704968939179 - 0.000000000501880286639411066538240515028660*I a51= -0.000000000537196806513684729782755840381852 - 0.000000000431608328308169134705317212956109*I a52= -1.20478708183463435222238448963980 E-10 + 0.000000000436782054534548353758126518996639*I a53=  0.000000000294236873163964332731548860943212 - 4.55731004522102747603632604275855 E-11*I a54= -1.09484348258184241455411404313358 E-10 - 1.62030719351912555496652240574897 E-10*I a55= -6.29394314688713941314843553884342 E-11 + 1.11742766424676835413587432779026 E-10*I a56=  8.36255959878877945937704291986788 E-11 + 1.10291562562161012484293879712436 E-12*I a57= -2.90513568874658489576233017176322 E-11 - 4.39013600472770972673122513274529 E-11*I a58= -2.29811153532266632771543167176271 E-11 + 3.63101011822347872513939835257048 E-11*I a59=  3.25899687388650229010887282083843 E-11 + 7.74390022108389232195175114220041 E-12*I a60= -1.81868183907415228508371813317456 E-12 - 2.47987652017774192289218548315937 E-11*I a61= -1.65748690876379100905877610466545 E-11 + 6.56094294458485238945177789424690 E-12*I a62=  7.89671428527872708294832099697662 E-12 + 9.65558082056524898977076618765452 E-12*I a63=  4.62125774179362854874547306781964 E-12 - 7.22876756098381830518401046879771 E-12*I a64= -5.65759620988165606459426764197572 E-12 - 1.40447397757024384653801223882907 E-12*I a65=  3.59230043053171442330785946914775 E-13 + 3.90929031426616411142571951897310 E-12*I a66=  2.38110114449910913864737055760666 E-12 - 1.11019536640239763020515574170534 E-12*I a67= -1.24446898293292890148742241252767 E-12 - 1.23110283238676648386146968644129 E-12*I a68= -4.69370002626149629187539175418096 E-13 + 1.06415995463525241744580231150137 E-12*I a69=  7.70901530570370360423583613971827 E-13 + 3.11212443861388366484406050529848 E-14*I a70= -1.73362371646043940990070178815011 E-13 - 4.81224041607396039360455113531847 E-13*I a71= -2.49580112873293178112472146938238 E-13 + 2.29709331425666375587118836029636 E-13*I a72=  2.06048029752544915803470623281604 E-13 + 9.07772793408009314804167345432811 E-14*I a73= -1.99981988952511503900017626279692 E-15 - 1.49659766697857426926842845765267 E-13*I a74= -8.91032922459351328104300575456857 E-14 + 4.45637690786538844143278688005410 E-14*I a75=  5.38920627291828173069098689493713 E-14 + 3.85014408140980596462765810389930 E-14*I a76=  8.39672845827494984430565262413313 E-16 - 4.28745581469085423500119831314564 E-14*I a77= -3.07833860695434498416121080170439 E-14 + 1.39064038624055069395977971825958 E-14*I a78=  1.82651480046632644960792775815767 E-14 + 1.84286959022402725882644514360893 E-14*I a79=  8.56674454279163246970950744512832 E-15 - 1.71479132326068121784553996632081 E-14*I a80= -1.35903799211189751496637159522325 E-14 - 1.97835940687435308751982365458644 E-15*I « Next Oldest | Next Newest »

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