03/08/2012, 09:51 PM
(This post was last modified: 03/09/2012, 12:10 AM by sheldonison.)
More on the Shell Thron boundary, itself. For a rational period of 5, or any other rational period, there is no superfunction. So the function becomes more and more chaotic as real(z) increases, or decreases. For increasing z, the chaos occurs around integer values+1/2, and for decreasing so, it occurs near the negative integers, where the singularity becomes stronger, instead of decaying. There are probably additional singularities, as well. Here I graph little pieces of the merged superfunction, to show the behavior. In my previous post, the merged sexp(z) was generated from a base with an irrationally indifferent fixed point. That sexp(z) solution does not show any visible misbehavior, in similar graphs. Presumably, the misbehavior would depend on the continued fraction representation of the period. I think the misbehavior may eventually extend to arbitrarily large values of imag(z). Since there is no superfunction, my algorithm cannot extend results to increasing imag(z), but Dimitrii's method should allow extending this rationally indifferent sexp(z) function as imag(z) goes to +i infinity. Here, I graph sexp(z)-L, so that the colors loop around the fixed point in a rainbow.
This image shows the behavior as real(z) decreases.
A couple of years ago, Henryk started a thread on new results from complex dynamics. It be interesting to know what complex dynamics says about this case, where one of the fixed points is rationally indifferent, and the other is a normal repelling fixed point, and/or the case where the one of the fixed points is irrationally indifferent.
- Sheldon
For reference, these are the taylor series coefficients used for the plots in this post, which is for the merged fixpoint solution for the base whose upper superfunction has a 'period'=5, with results accurate to approximately 26 decimal digits.
This image shows the behavior as real(z) decreases.
A couple of years ago, Henryk started a thread on new results from complex dynamics. It be interesting to know what complex dynamics says about this case, where one of the fixed points is rationally indifferent, and the other is a normal repelling fixed point, and/or the case where the one of the fixed points is irrationally indifferent.
- Sheldon
For reference, these are the taylor series coefficients used for the plots in this post, which is for the merged fixpoint solution for the base whose upper superfunction has a 'period'=5, with results accurate to approximately 26 decimal digits.
Code:
B = 1.965138765983644624714311722 + 0.4412433260153932508689331862*I
a0= 1
a1= 0.9079807434839744851746261607 + 0.1591412728333623157413881350*I
a2= 0.007659492127868240456307876222 + 0.1846399330316959146545308260*I
a3= 0.06959571260339947521384981633 + 0.05494395086384639072904240777*I
a4= -0.01853896739586610864156284686 + 0.03208183862752477800053505024*I
a5= 0.004343316287399599447846087888 + 0.006508307197205574089532061575*I
a6= -0.005345207961363676870744658158 + 0.003585318340289147472698483962*I
a7= 0.0004398010032873102933211473086 + 0.0001619039523931708562962812843*I
a8= -0.001003777877024937424502267473 + 0.0002862394068353525818737587700*I
a9= 0.0001479472026338298193569454915 - 0.00009953065219165022072959024958*I
a10= -0.0001662244954792588969007927661 + 0.00002235001201454921438165448298*I
a11= 0.00004655824195004755206712949921 - 0.00002767430137890164060186821929*I
a12= -0.00002876332027277035001458540190 + 0.000004231641397656991640456123169*I
a13= 0.00001183530971334214823943042012 - 0.000005341504387449057331084908997*I
a14= -0.000005612719915424328356904312019 + 0.000001289094648662858844870211635*I
a15= 0.000002702128429420899103644987746 - 0.0000009799472019399010609484192961*I
a16= -0.000001205336226991155083066773933 + 0.0000003518138615910355943883918922*I
a17= 0.0000005963874917370866392729873832 - 0.0000001924208068631061932662511999*I
a18= -0.0000002703191296883001291959068737 + 0.00000008552425286430910256950860007*I
a19= 0.0000001319738521258292228657432965 - 0.00000004105838543838365584367414149*I
a20= -0.00000006146589583669697426749838905 + 0.00000001970691268589321217243576770*I
a21= 0.00000002961711704252429566540478908 - 0.000000009210492290507201311794151223*I
a22= -0.00000001404873356874825470800388787 + 0.000000004479659273013273766147094565*I
a23= 0.000000006737179859607704090354177211 - 0.000000002109766520357551734717742807*I
a24= -0.000000003225884711771446753906410292 + 0.000000001022237304639772309102919065*I
a25= 0.000000001548114388103841631193419049 - 0.0000000004871410509379935244138690376*I
a26= -0.0000000007446997429243798877475250267 + 0.0000000002352112954763236930800349863*I
a27= 0.0000000003583501502122949980151043462 - 1.129923889224470533764393227 E-10*I
a28= -1.728603213753762812740473800 E-10 + 5.453507804323909583206114089 E-11*I
a29= 8.342050243813429497490025940 E-11 - 2.631831742274014885336360246 E-11*I
a30= -4.032925253281005910200771743 E-11 + 1.272055494611321230410440559 E-11*I
a31= 1.951147105623435138879360524 E-11 - 6.155918034173348886504797412 E-12*I
a32= -9.451578356062999050506602176 E-12 + 2.981306712035548668200384191 E-12*I
a33= 4.582421795355821824849509605 E-12 - 1.445678022921137883431517914 E-12*I
a34= -2.223852478904982341991740882 E-12 + 7.015089710943148331012188469 E-13*I
a35= 1.080154524081000844565622534 E-12 - 3.407556284021761025254119328 E-13*I
a36= -5.250739238722500796005850588 E-13 + 1.656384186376948876500405934 E-13*I
a37= 2.554423001336741090860955710 E-13 - 8.058265964053152649733211913 E-14*I
a38= -1.243596553139020331371677062 E-13 + 3.923059787337772543123214696 E-14*I
a39= 6.058562914071990590596928595 E-14 - 1.911243134624896486672673790 E-14*I
a40= -2.953544318944044594994858795 E-14 + 9.317298427938769206399126597 E-15*I
a41= 1.440754893894594649211638307 E-14 - 4.545022731062749478616983881 E-15*I
a42= -7.032251751204233617915117501 E-15 + 2.218405535233745344509350613 E-15*I
a43= 3.434356605653919441915312888 E-15 - 1.083406554742385931896859377 E-15*I
a44= -1.678151288311960576191600926 E-15 + 5.293921578047610160999371309 E-16*I
a45= 8.204295608740320516285620802 E-16 - 2.588138352126890810641316895 E-16*I
a46= -4.012970658337325798408519088 E-16 + 1.265937579047528306560052950 E-16*I
a47= 1.963794131970996574622400360 E-16 - 6.195012716053768520438385859 E-17*I
a48= -9.614408915812041114031088261 E-17 + 3.032975232054436438030567546 E-17*I
a49= 4.709098185791310260074042045 E-17 - 1.485538829966010884395285567 E-17*I
a50= -2.307458133217148892201936376 E-17 + 7.279140385851274361399816673 E-18*I
a51= 1.131106921186611173052383837 E-17 - 3.568206057029077684947420636 E-18*I
a52= -5.546774346291304158250684900 E-18 + 1.749793353533065003341975364 E-18*I
a53= 2.721059107919533582186112050 E-18 - 8.583891945510160809496403880 E-19*I
a54= -1.335334563689921674928873757 E-18 + 4.212465482148494948392331835 E-19*I
a55= 6.555278763677110023492727087 E-19 - 2.067937603451532486240278374 E-19*I
a56= -3.219110105500072144843864097 E-19 + 1.015505070445530642909563527 E-19*I
a57= 1.581317244201818746662513402 E-19 - 4.988445956813825095210712446 E-20*I
a58= -7.770265770670810750547222828 E-20 + 2.451219139803678807240155259 E-20*I
a59= 3.819283189576525330297209993 E-20 - 1.204836536477911371846248647 E-20*I
a60= -1.877814232551506691213926782 E-20 + 5.923779651999245740383199924 E-21*I
a61= 9.235151938298962165893936310 E-21 - 2.913334225847860373679672491 E-21*I
a62= -4.543098915097051302883381738 E-21 + 1.433172483793440107934695348 E-21*I
a63= 2.235493254326134998753514027 E-21 - 7.052119798843451395387359767 E-22*I
a64= -1.100281930305726854199048095 E-21 + 3.470965941697627634793739024 E-22*I
a65= 5.416772282846451016308228898 E-22 - 1.708783103833573008616087752 E-22*I
a66= -2.667348397740396154799836426 E-22 + 8.414446178301270078080072237 E-23*I
a67= 1.313767413079105214069445223 E-22 - 4.144421630153321600025135107 E-23*I
a68= -6.472241541363306324329732163 E-23 + 2.041737620118049490106781936 E-23*I
a69= 3.189228872612451515471409849 E-23 - 1.006084093575876170617783093 E-23*I
a70= -1.571839954555664790501461317 E-23 + 4.958550391145711493488891033 E-24*I
a71= 7.748421558917525267188408130 E-24 - 2.444317288792335780386575226 E-24*I
a72= -3.820431440274657698257468206 E-24 + 1.205155001518322479813597008 E-24*I
a73= 1.884134025252691322201483888 E-24 - 5.944277070332361533838162673 E-25*I
a74= -9.295301766583011369912095735 E-25 + 2.933067459410344555731401738 E-25*I
a75= 4.585498801958408021086081911 E-25 - 1.446978970199893842357274206 E-25*I
a76= -2.261579575434027128422136034 E-25 + 7.126893696589671916611800212 E-26*I
a77= 1.114376323355299788740067499 E-25 - 3.503734579154330225494917534 E-26*I
a78= -5.502806693794137845693412775 E-26 + 1.729697113477494149082120615 E-26*I
a79= 2.722297695938104032335612083 E-26 - 8.585723035060422857377695280 E-27*I
a80= -1.347403026381502750265909122 E-26 + 4.265578796441743424478689569 E-27*I
a81= 6.555286514914951073829086733 E-27 - 2.007571996524703178872730421 E-27*I