NumberDB$^\beta$ Advanced search Tags Tables Help Login
Periodic windows of the logistic map by kneading word
edit · history · discussion · files · short url · bifurcation theory dynamical systems
Numbers
$W$
quantity 
value
$\mathtt{RLC}$
$r_{\mathrm{on}}$:
3.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145
comment: period $3$; $r_{\mathrm{on}}=1+2\sqrt2$ [8].
$\mathtt{RLC}$
$c_{\mathrm{on}}$:
-7/4
comment: period $3$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLC}$
$r_W$:
3.831874055283315568410362775496106555797827852603694630478890447774503731227095794442737177288907762
comment: period $3$; $r_W$ is the root near $3.83187$ of $r^6-6r^5+4r^4+24r^3-16r^2-32r-64$.
$\mathtt{RLC}$
$c_W$:
-1.754877666246692760049508896358528691894606617772793143989283970646080655128081090738227092842250304
comment: period $3$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLC}$
$r_{\mathrm{pd}}$:
3.841499007543507846310702443843943153164514952031854751184826450165563461320098490290154875325010576
comment: period $3$; $r_{\mathrm{pd}}$ is the root near $3.84150$ of $r^6-6r^5+4r^4+24r^3-14r^2-36r-81$ [9].
$\mathtt{RLC}$
$c_{\mathrm{pd}}$:
-1.768529152467685015118134998577843403365121825731680418685500565103535794805538536005749121977272489
comment: period $3$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLC}$
$h_{\mathrm{top}}$:
0.4812118250596034474977589134243684231351843343856605196610181688401638676082217744120094291227234750
comment: period $3$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^2 - x - 1$; $h_{\mathrm{top}}=\log\lambda$; $\lambda$ is the golden ratio $\varphi$.
$\mathtt{RLLC}$
$r_{\mathrm{on}}$:
3.960101882689952012229267441585598365547366031312451954555278215550241993899863080714046307957359072
comment: period $4$.
$\mathtt{RLLC}$
$c_{\mathrm{on}}$:
-1.940550788976149606063779229454231195293619995924889757356214321368912379218164379955158159946665718
comment: period $4$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLC}$
$r_W$:
3.960270127221152601571257798501420544549781431863804269630105765676572279086869849589322270677607577
comment: period $4$.
$\mathtt{RLLC}$
$c_W$:
-1.940799806529484752232090979655204176869040714461035523938210176681262797645454057183436754362102789
comment: period $4$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLC}$
$r_{\mathrm{pd}}$:
3.960768652406645256046371427380866877425475433389218910176759920822075912075442552844418007072453943
comment: period $4$.
$\mathtt{RLLC}$
$c_{\mathrm{pd}}$:
-1.941537753268465539334645081335137302112963652443093023862503797514245782358240271484061333910837329
comment: period $4$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLC}$
$h_{\mathrm{top}}$:
0.6093778634360062315368033711683986954285392793128541477762892366225152051203195768397057073419585497
comment: period $4$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^3 - x^2 - x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLRRC}$
$r_{\mathrm{on}}$:
3.738172375265962369430261559679531973444115440489991922904288455334944230717097796051709049574841765
comment: period $5$; $r_{\mathrm{on}}$ is the first period-$5$ onset listed in OEIS A118452 [10].
$\mathtt{RLRRC}$
$c_{\mathrm{on}}$:
-1.624396989167410562649427341357693883454238107399013052973571375151839118658085300175357732192165002
comment: period $5$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRC}$
$r_W$:
3.738914912970684993085212890376289451230116550807871367006256801980091017452887955021757582632163057
comment: period $5$.
$\mathtt{RLRRC}$
$c_W$:
-1.625413725123303737443410575023330183802255938091052410634006738178386457316026119858018767879139791
comment: period $5$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRC}$
$r_{\mathrm{pd}}$:
3.741120756632440206307293823670998371000508432656225255249811565073090684557011894475098622922002505
comment: period $5$.
$\mathtt{RLRRC}$
$c_{\mathrm{pd}}$:
-1.628435750610300372318917760766091968591955392160702279203567325940312210802218870638490604481654313
comment: period $5$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRC}$
$h_{\mathrm{top}}$:
0.4140127373937918204806368153100175578546101763083410076870680620346485273270767355612604258375425856
comment: period $5$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^4 - x^3 - x^2 + x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLRC}$
$r_{\mathrm{on}}$:
3.905571870172499788433165533744113866118423918706080014361586222258367784574314390881789615335012690
comment: period $5$.
$\mathtt{RLLRC}$
$c_{\mathrm{on}}$:
-1.860586973184429491433617619202588231152110652446376460387939716751506781700242161445100609535816259
comment: period $5$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRC}$
$r_W$:
3.905706469831290354820407092052706925201235915074035546540387773832967560115483920668424640039078843
comment: period $5$.
$\mathtt{RLLRC}$
$c_W$:
-1.860782522204854871232242023799874080915734775039867996915880375880518999473389628554990125469320812
comment: period $5$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRC}$
$r_{\mathrm{pd}}$:
3.906107428137990139409734699674818990931164746663857124630803191488353481716595547315907674321835483
comment: period $5$.
$\mathtt{RLLRC}$
$c_{\mathrm{pd}}$:
-1.861365095969700880569453092760749831422857648194405814350349597874379871110916741802428475276489024
comment: period $5$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRC}$
$h_{\mathrm{top}}$:
0.5435350724978695498926364006192337217181767537693045902578565450006843770349604030842035014653078917
comment: period $5$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^4 - x^3 - x^2 - x + 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLC}$
$r_{\mathrm{on}}$:
3.990257307413383568835334751017794362481470848315599601899881892646826819868813562370790452724779994
comment: period $5$.
$\mathtt{RLLLC}$
$c_{\mathrm{on}}$:
-1.985409691134784680944520103256752125816322197232533177342461889138223687794624848575549797820489719
comment: period $5$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLC}$
$r_W$:
3.990267046973701518277526388554317983365989821566078137012259342089850199944116012482342685236159997
comment: period $5$.
$\mathtt{RLLLC}$
$c_W$:
-1.985424253054205310609750582718674337261701607315313871640456439268917891459381820804266931996038787
comment: period $5$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLC}$
$r_{\mathrm{pd}}$:
3.990296185577653667268262304717267208417954589169608982726031362737549971787550078904182461935827168
comment: period $5$.
$\mathtt{RLLLC}$
$c_{\mathrm{pd}}$:
-1.985467819370066335075193010514793892765796313362359379115885207603067007871259199091137302447081533
comment: period $5$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLC}$
$h_{\mathrm{top}}$:
0.6562559792369758084816619843129949085508630230767612918556264734285532085454909658008270936824151612
comment: period $5$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^4 - x^3 - x^2 - x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLRRRC}$
$r_{\mathrm{on}}$:
3.626553161694973725877232252093317491709475785795024696802437226790072120100839389126324793567313628
comment: period $6$; $r_{\mathrm{on}}$ is the first period-$6$ onset listed in OEIS A118453 [11].
$\mathtt{RLRRRC}$
$c_{\mathrm{on}}$:
-1.474695377802465698628143054748346551819157517855114442759213670487618182614369924218623943133298054
comment: period $6$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRRC}$
$r_W$:
3.627557529515523203374874102017441679844714136686374286012689587697194022095545243782090031824496942
comment: period $6$.
$\mathtt{RLRRRC}$
$c_W$:
-1.476014642728429897517365371395994438779215886771978538678946105805340164802932069582523021733869361
comment: period $6$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRRC}$
$r_{\mathrm{pd}}$:
3.630388700012403939680192121843394176179858556711454807750726264145749087549006701107546781562950989
comment: period $6$.
$\mathtt{RLRRRC}$
$c_{\mathrm{pd}}$:
-1.479736178288236091385167949992841066494132902697659749912371122707702333320187333031930698077303512
comment: period $6$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRRC}$
$h_{\mathrm{top}}$:
0.2406059125298017237488794567121842115675921671928302598305090844200819338041108872060047145613617375
comment: period $6$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^4 - x^2 - 1$; $h_{\mathrm{top}}=\log\lambda$; $h_{\mathrm{top}}$ is half the entropy of $\mathtt{RLC}$.
$\mathtt{RLLRLC}$
$r_{\mathrm{on}}$:
3.841499007543507846310702443843943153164514952031854751184826450165563461320098490290154875325010576
comment: period $6$; $r_{\mathrm{on}}$ is the first period-doubling point of $\mathtt{RLC}$.
$\mathtt{RLLRLC}$
$c_{\mathrm{on}}$:
-1.768529152467685015118134998577843403365121825731680418685500565103535794805538536005749121977272489
comment: period $6$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLC}$
$r_W$:
3.844568792194432972900784876245583103531857468454920780151622672757676973037934822849180481222279512
comment: period $6$.
$\mathtt{RLLRLC}$
$c_W$:
-1.772892903381623799434128230874643248588007176944030043908240150268782170722308166776912159525908237
comment: period $6$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLC}$
$r_{\mathrm{pd}}$:
3.847610661179367621802588375816176703968577555425276856248017121922341893535302910547935569489533356
comment: period $6$.
$\mathtt{RLLRLC}$
$c_{\mathrm{pd}}$:
-1.777221619415598806299676534122188773457619027519213013491557243394457166990805543410018346698564709
comment: period $6$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLC}$
$h_{\mathrm{top}}$:
0.4812118250596034474977589134243684231351843343856605196610181688401638676082217744120094291227234750
comment: period $6$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^2 - x - 1$; $h_{\mathrm{top}}=\log\lambda$; the entropy equals that of $\mathtt{RLC}$.
$\mathtt{RLLRRC}$
$r_{\mathrm{on}}$:
3.937516418983030658919905029393234867268426368582458396785601151276292369365133664892565150033925239
comment: period $6$.
$\mathtt{RLLRRC}$
$c_{\mathrm{on}}$:
-1.907250677948722031228876904515825664748053457302999457823758076358302303057042951379629921318480427
comment: period $6$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRC}$
$r_W$:
3.937536444754551104561950581971038733652159155152240382840687368153059815867785081333482369794379584
comment: period $6$.
$\mathtt{RLLRRC}$
$c_W$:
-1.907280091065301968397929087482382736408205369170374848433827429587273280008659144640076682537169035
comment: period $6$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRC}$
$r_{\mathrm{pd}}$:
3.937596466686345177756987847219078465183006263187776363685746501386819776260086853926648991906034873
comment: period $6$.
$\mathtt{RLLRRC}$
$c_{\mathrm{pd}}$:
-1.907368250272024873435302079155424224935622954688941975834124334363798479298640665563717574632933379
comment: period $6$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRC}$
$h_{\mathrm{top}}$:
0.5835568193858775853074276352840092863922199799502718224832316530257187332361298868272713897349969521
comment: period $6$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^5 - x^4 - x^3 - x^2 + x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLRC}$
$r_{\mathrm{on}}$:
3.977760440936972441070420337022955798631105365769945880301537870542484270889914018942344874599547837
comment: period $6$.
$\mathtt{RLLLRC}$
$c_{\mathrm{on}}$:
-1.966764310902288134414343805710578152305616395636589787986264420426350269324718492010280087630485328
comment: period $6$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRC}$
$r_W$:
3.977766422265472914020545451260876758642722685640425884501434428015535816949087140326017511526655430
comment: period $6$.
$\mathtt{RLLLRC}$
$c_W$:
-1.966773216392928685678055598464378166491189446526618286047326346691721175596583632444302783310915776
comment: period $6$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRC}$
$r_{\mathrm{pd}}$:
3.977784350737002120279235567433295914983496745761000918216360211427166147740260969168118706730441919
comment: period $6$.
$\mathtt{RLLLRC}$
$c_{\mathrm{pd}}$:
-1.966799909873547314977955682657141324886757708396555808166169613006915685918022035285101526151347576
comment: period $6$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRC}$
$h_{\mathrm{top}}$:
0.6329743192009474501481832460801859300728576976886075268319739483062799438148109826273510874461527725
comment: period $6$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^4 - 2*x^3 + x^2 - 2*x + 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLLC}$
$r_{\mathrm{on}}$:
3.997582523904762524251454978910488316042522324828258602884655653466709476647198529876987852577062785
comment: period $6$.
$\mathtt{RLLLLC}$
$c_{\mathrm{on}}$:
-1.996375246904811547535411357374172843493900213923548108181814138427380024055195214312000209273524183
comment: period $6$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLC}$
$r_W$:
3.997583118254567266100194720817863093675017299222944348764191247141084349032453845784339704645429475
comment: period $6$.
$\mathtt{RLLLLC}$
$c_W$:
-1.996376137711193750644879819060606618293370308633844154239242430988582520429178934025958304373872323
comment: period $6$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLC}$
$r_{\mathrm{pd}}$:
3.997584900125419618837829701416315498873057274159621308775943153115203052215595059385017808317304530
comment: period $6$.
$\mathtt{RLLLLC}$
$c_{\mathrm{pd}}$:
-1.996378808364980477799949451065884598254755577489927744625682182436925645634590238965968536579852405
comment: period $6$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLC}$
$h_{\mathrm{top}}$:
0.6759746921034222048702355102771254108805217604721743026195707245561942310098481569582292290018313319
comment: period $6$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^5 - x^4 - x^3 - x^2 - x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLRRRRC}$
$r_{\mathrm{on}}$:
3.701640764160349581824643789840889220144291589515206443123456257307919373552959778240516280242008702
comment: period $7$; $r_{\mathrm{on}}$ is the first period-$7$ onset listed in OEIS A118746 [12].
$\mathtt{RLRRRRC}$
$c_{\mathrm{on}}$:
-1.574715704643229407380333578304632229082622726657881487010477669486692189146062413789805022828606714
comment: period $7$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRRRC}$
$r_W$:
3.701769153537956114193310843013006048494133215749332831848494410468474023169676359504177384520527374
comment: period $7$.
$\mathtt{RLRRRRC}$
$c_W$:
-1.574889139752300969819965552495974283771948275051920108329908603574997581722351888558432214297115957
comment: period $7$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRRRC}$
$r_{\mathrm{pd}}$:
3.702154928153588770222612312426413655918603425946700817575042789935462662015847094896913198844497126
comment: period $7$.
$\mathtt{RLRRRRC}$
$c_{\mathrm{pd}}$:
-1.575410313936181622177143136112878249492078662022609900882148922101187426988858449035375580650423930
comment: period $7$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRRRC}$
$h_{\mathrm{top}}$:
0.3822450858400356413293584991848573937594164224201954300292839383616548905505831820170135085159009127
comment: period $7$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^3 - x^2 - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLRRLRC}$
$r_{\mathrm{on}}$:
3.774133385584886599999544750968183948302478327288125754512327632928741133732183548824565013894318087
comment: period $7$.
$\mathtt{RLRRLRC}$
$c_{\mathrm{on}}$:
-1.673954010254166278084462122945931947374323563645999487409213474555793083156599017169805046622236050
comment: period $7$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRLRC}$
$r_W$:
3.774214188900913232127399722041048182060840120285006726450249002712212424365581891999803919920051863
comment: period $7$.
$\mathtt{RLRRLRC}$
$c_W$:
-1.674066091474787971565296029172325596206403719925258163289474362780555130416606811320252923549433643
comment: period $7$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRLRC}$
$r_{\mathrm{pd}}$:
3.774455772798689443686290958262552223210350194122253711520712260490572313597387890407678478565346325
comment: period $7$.
$\mathtt{RLRRLRC}$
$c_{\mathrm{pd}}$:
-1.674401208803993264694925660075453787531770094417279825391017812312196254867080563235312547982857297
comment: period $7$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRLRC}$
$h_{\mathrm{top}}$:
0.4421378287316384595584503617785560502498667018373349568907998184867266671127156086440823217405674868
comment: period $7$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^6 - x^5 - x^4 + x^3 - x^2 - x + 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLRLRC}$
$r_{\mathrm{on}}$:
3.886028805002814483072083236371992144949475335183289449896090762441557076796503452148438692021724613
comment: period $7$.
$\mathtt{RLLRLRC}$
$c_{\mathrm{on}}$:
-1.832290565826493345858610536092337273551936518513513693443121591977904821318763577237569326824504995
comment: period $7$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLRC}$
$r_W$:
3.886045878187822011522491985166861831306189197736503862108247851787295523774191923400144994632359944
comment: period $7$.
$\mathtt{RLLRLRC}$
$c_W$:
-1.832315202751229192084897526042518143229339893506219597603881639013923985476096050081996383480503411
comment: period $7$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLRC}$
$r_{\mathrm{pd}}$:
3.886097052252474703589062170547314269036143467080959326548151736528028632100737755207381864626880528
comment: period $7$.
$\mathtt{RLLRLRC}$
$c_{\mathrm{pd}}$:
-1.832389048755105924882418943918758669806857089849467378469676239731433128849883473268735723520898397
comment: period $7$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLRC}$
$h_{\mathrm{top}}$:
0.5223150571797636676057540713298795817891139228335266307655807783463452712885502313376428095700000838
comment: period $7$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^6 - x^5 - x^4 - x^3 + x^2 + x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLRRRC}$
$r_{\mathrm{on}}$:
3.922185905463235174179160705168870316344728999094299495391706381213661188929578277575057402605557724
comment: period $7$.
$\mathtt{RLLRRRC}$
$c_{\mathrm{on}}$:
-1.884792616521996904646928385336366080822185910855904385302351886309935152770167914879902727195268095
comment: period $7$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRRC}$
$r_W$:
3.922193403309700028472413580359373772802740342844941601446308625996417829894533595664919275066297848
comment: period $7$.
$\mathtt{RLLRRRC}$
$c_W$:
-1.884803571586681792329478092915839649635927126568451086477445146339028585107127064035334694409711173
comment: period $7$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRRC}$
$r_{\mathrm{pd}}$:
3.922215881776105533615917879334243503629088834836199613785118545143504746592023019621926940928473553
comment: period $7$.
$\mathtt{RLLRRRC}$
$c_{\mathrm{pd}}$:
-1.884836414926125498232792240518496196070307341647882919967227080375538154765870077981761703059122389
comment: period $7$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRRC}$
$h_{\mathrm{top}}$:
0.5623991486459236930241015281357565995840464514881329253514606686360769061242417826459754149509881096
comment: period $7$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^3 - 2*x^2 + x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLRRLC}$
$r_{\mathrm{on}}$:
3.951027355414705409366713613009604520554202082306043298818935927181239165902425268297578122836472343
comment: period $7$.
$\mathtt{RLLRRLC}$
$c_{\mathrm{on}}$:
-1.927140613101477509946817693360616788661682652845526765387054396927325708491690436342219103996105579
comment: period $7$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRLC}$
$r_W$:
3.951032164761306197891492318692053579500978056252822261505773604818792971458708401105986848757068516
comment: period $7$.
$\mathtt{RLLRRLC}$
$c_W$:
-1.927147709363950262460068188946594278007127957666936791637019227516202318233430701742416720301058406
comment: period $7$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRLC}$
$r_{\mathrm{pd}}$:
3.951046597417378452087715599473224682648629170776503276640591936743237100601604618080761910729196353
comment: period $7$.
$\mathtt{RLLRRLC}$
$c_{\mathrm{pd}}$:
-1.927169005032171732640841525731421722737437803911240766211194608473214653369536384821047768600867005
comment: period $7$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRLC}$
$h_{\mathrm{top}}$:
0.6010014635615238945922165119986414954352882766962744088188349494158573627662994406742439999919567061
comment: period $7$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^6 - x^5 - x^4 - x^3 + x^2 - x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLRLC}$
$r_{\mathrm{on}}$:
3.968974213355921489848012727002553905694225522756762058031468133763672907555006277463179724045687631
comment: period $7$.
$\mathtt{RLLLRLC}$
$c_{\mathrm{on}}$:
-1.953701969893103204887315564617102922246925862783939778732946421555976947509416171380520255452039417
comment: period $7$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRLC}$
$r_W$:
3.968976856955537971495294805539108946037783703799351121542877785508392114874854378195569440319125445
comment: period $7$.
$\mathtt{RLLLRLC}$
$c_W$:
-1.953705894284396245427622199013653238901535456878068475145301474330533287552533662410171083425556907
comment: period $7$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRLC}$
$r_{\mathrm{pd}}$:
3.968984782564841919744772307183367285778950551696867634862611772963580809917611820441102217769464538
comment: period $7$.
$\mathtt{RLLLRLC}$
$c_{\mathrm{pd}}$:
-1.953717659775400413058708750616463327489601380636663062688361755520433966886457605428694554456386377
comment: period $7$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRLC}$
$h_{\mathrm{top}}$:
0.6183622093451163254328878318858700802410097601306338174398081638756018957724383667073654105154595007
comment: period $7$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^6 - x^5 - x^4 - x^3 - x^2 + x + 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLRRC}$
$r_{\mathrm{on}}$:
3.984746617117961562539941995044746516855226230365388320399580625808205835600273687798270302880259136
comment: period $7$.
$\mathtt{RLLLRRC}$
$c_{\mathrm{on}}$:
-1.977178092099278859617041631102235126538406603940753906641173250247915562322733886927312594866196777
comment: period $7$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRRC}$
$r_W$:
3.984747618815538907772276907561454679676957411498679554095395415385117352738851840883073537397452987
comment: period $7$.
$\mathtt{RLLLRRC}$
$c_W$:
-1.977179587006257387346088520662828616836483119614126188894964036958779814776736129422816867913153305
comment: period $7$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRRC}$
$r_{\mathrm{pd}}$:
3.984750623369319488663335684641902270522050587697569427282012348360632989626388514305925150107921745
comment: period $7$.
$\mathtt{RLLLRRC}$
$c_{\mathrm{pd}}$:
-1.977184070925885319221065532173161600744944476011374154654110859657494742592858244654120427462584618
comment: period $7$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRRC}$
$h_{\mathrm{top}}$:
0.6457107068710058949508749656015280172218090248209520030562287850906054488818234514874051685448907444
comment: period $7$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^6 - x^5 - x^4 - x^3 - x^2 + x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLLRC}$
$r_{\mathrm{on}}$:
3.994537466821304094471459652738996432157484347436955481659634402941910775933929920030709206742374548
comment: period $7$.
$\mathtt{RLLLLRC}$
$c_{\mathrm{on}}$:
-1.991813660049138230105629802816033248198703637985623481551004033231855948014811987199475026549790230
comment: period $7$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLRC}$
$r_W$:
3.994537809111197190417220573536005876265512277401941519589583013418430887712192292375342353360909627
comment: period $7$.
$\mathtt{RLLLLRC}$
$c_W$:
-1.991814172549122215732560949862288112398840479094439836523016087980045018029971370721309604046086354
comment: period $7$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLRC}$
$r_{\mathrm{pd}}$:
3.994538835800214273101423989020183665537761309219249903005457208334390054246504862911868583126934979
comment: period $7$.
$\mathtt{RLLLLRC}$
$c_{\mathrm{pd}}$:
-1.991815709778925664971840735179586886714583533218322676297960179157499022994670938220731244133427261
comment: period $7$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLRC}$
$h_{\mathrm{top}}$:
0.6662159015140127416264159553911686576217155658796753187985835509073395889042876179832187252017783508
comment: period $7$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^6 - x^5 - x^4 - x^3 - x^2 - x + 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLLLC}$
$r_{\mathrm{on}}$:
3.999397024083989298851031655198098311928961334225407147109489112991623382006088004782805101637356311
comment: period $7$.
$\mathtt{RLLLLLC}$
$c_{\mathrm{on}}$:
-1.999095627020972770512579033545857080458572046257630738666249422504396024167972927284553213425134487
comment: period $7$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLLC}$
$r_W$:
3.999397060962098411191495707485953442874478379741847608438667159554951228082454648646320743388701683
comment: period $7$.
$\mathtt{RLLLLLC}$
$c_W$:
-1.999095682327018473210629999222294462129240158216774560026431683438161266678158911694297434848587857
comment: period $7$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLLC}$
$r_{\mathrm{pd}}$:
3.999397171578339685885389178679490160210002792117128770079994100321368536765179901195306601142436685
comment: period $7$.
$\mathtt{RLLLLLC}$
$c_{\mathrm{pd}}$:
-1.999095848218036019194451664450140806924040597328020379450530301297250672551525444536043315172533637
comment: period $7$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLLC}$
$h_{\mathrm{top}}$:
0.6849047263840670390265450045421892791553012853281446507991516039400347971679896430573147260920963341
comment: period $7$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^6 - x^5 - x^4 - x^3 - x^2 - x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLRRRRRC}$
$r_{\mathrm{on}}$:
3.662108913209440482494985179401963324650410704107152165354024618133906761088718265064650386993431985
comment: period $8$.
$\mathtt{RLRRRRRC}$
$c_{\mathrm{on}}$:
-1.521705966447287079857426590871368336465139889444578447293332988746392859707338851152741447169990168
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRRRRC}$
$r_W$:
3.662192503686576753408760104323259108936150156636422138356293900513767325702574707143942294185756894
comment: period $8$.
$\mathtt{RLRRRRRC}$
$c_W$:
-1.521817231671250995197287331707767489051013082756997403689583385621888673294580368952335546937639231
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRRRRC}$
$r_{\mathrm{pd}}$:
3.662440707218951929165078336760059594086303533738695742749539226376857535483402474794235903432540537
comment: period $8$.
$\mathtt{RLRRRRRC}$
$c_{\mathrm{pd}}$:
-1.522147629864138226804627530138195947912311543148869209558512538819977500764194424328823656868599160
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRRRRC}$
$h_{\mathrm{top}}$:
0.3046889317180031157684016855841993477142696396564270738881446183112576025601597884198528536709792749
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^6 - x^4 - x^2 - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLRRLRRC}$
$r_{\mathrm{on}}$:
3.800740116185650269544019660574655861309171054123192705182714339639571202916446745946604193102552789
comment: period $8$.
$\mathtt{RLRRLRRC}$
$c_{\mathrm{on}}$:
-1.711036299602902442737690951277512930319039950384594600110155258433719086394963560881882838269175844
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRLRRC}$
$r_W$:
3.800770943874669769373681094541042531243472703358879522602247670292807561204709814947813850209126113
comment: period $8$.
$\mathtt{RLRRLRRC}$
$c_W$:
-1.711079470013152149832420460633546039822793809005903801634378634018166805386594484770160438100675051
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRLRRC}$
$r_{\mathrm{pd}}$:
3.800863743732897904084706007163771486772100384137176509039574256518946369845106365626002988508651811
comment: period $8$.
$\mathtt{RLRRLRRC}$
$c_{\mathrm{pd}}$:
-1.711209427739366095820057162592591641773652365550620286495157645838218896700024271390032733748352196
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLRRLRRC}$
$h_{\mathrm{top}}$:
0.4682578962892690238036677941634449344169555421683316556539680294062710052381178192465630827549336519
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^7 - x^6 - x^5 + x^4 - x^3 - x^2 + x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLRLRRC}$
$r_{\mathrm{on}}$:
3.870532112458717263207972364869970022767695740237793392956847349422364764798557724149372990811657125
comment: period $8$.
$\mathtt{RLLRLRRC}$
$c_{\mathrm{on}}$:
-1.809988652164176453236225785775167876673228995304522747952629963681413768981446612845349671697059530
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLRRC}$
$r_W$:
3.870540984363757223187552694640279443822824350492893990073711037436607789532630305195266521527276241
comment: period $8$.
$\mathtt{RLLRLRRC}$
$c_W$:
-1.810001385728012072726032393667559396296122986892711420916360880856159395711565821698619522407255262
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLRRC}$
$r_{\mathrm{pd}}$:
3.870567556387604950796347735272878721876250891292225812705573427397501939750073298042576013103312553
comment: period $8$.
$\mathtt{RLLRLRRC}$
$c_{\mathrm{pd}}$:
-1.810039523946276382188045485554661020085427753272787965259104514292073564586411288754173646589688744
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLRRC}$
$h_{\mathrm{top}}$:
0.4997469349065763168991860640637229310389301065375144476786707038210553682691211962756263081149196064
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^5 - x^4 - 2*x^2 + x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLRLRLC}$
$r_{\mathrm{on}}$:
3.899462408673439157018516146563113876052217522856198864337870406379532456949432946795493024496163428
comment: period $8$.
$\mathtt{RLLRLRLC}$
$c_{\mathrm{on}}$:
-1.851720564827595376038578538001835952834549716192905305197505615737715182028694607878339480789497874
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLRLC}$
$r_W$:
3.899468950970602334169343447876952058563575863402143335153498044409864620831012622634717790874904141
comment: period $8$.
$\mathtt{RLLRLRLC}$
$c_W$:
-1.851730049410641290596284907276327419107492715710647186733310413609309893471554769608371185572932297
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLRLC}$
$r_{\mathrm{pd}}$:
3.899488629309009671280936948826224897212120914987800847214076147843720205710519899310334651748989133
comment: period $8$.
$\mathtt{RLLRLRLC}$
$c_{\mathrm{pd}}$:
-1.851758577873059924338924018030085170565618385123812492346045088537181327718652331029802797532594083
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRLRLC}$
$h_{\mathrm{top}}$:
0.5397924974635598516522691635593408071246983296560781743694898196447683749373723518833851037497626782
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^7 - x^6 - x^5 - x^4 + x^3 + x^2 - x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLRRRLC}$
$r_{\mathrm{on}}$:
3.912042017282251570438911869551972263270073203430034409588531263973526045008764119452992536679421316
comment: period $8$.
$\mathtt{RLLRRRLC}$
$c_{\mathrm{on}}$:
-1.869997177604321288511089735201491105115020084787641987260505945231724668608092707299109609110054444
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRRLC}$
$r_W$:
3.912046621075126511047630238514592355234828476629003000739736565922825911070253189226716469193009517
comment: period $8$.
$\mathtt{RLLRRRLC}$
$c_W$:
-1.870003880828765361573296401227468790696423696464304693260405238171948285162690925062060734303712142
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRRLC}$
$r_{\mathrm{pd}}$:
3.912060397754645243588899978404765138337402146939599270071460195423799135989730090654032567636106223
comment: period $8$.
$\mathtt{RLLRRRLC}$
$c_{\mathrm{pd}}$:
-1.870023940042735666212199248871173443708967774213352833867571341028878363918626515391419157296956499
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRRLC}$
$h_{\mathrm{top}}$:
0.5476118702402008170801971585567613545872935585570976556005129166067851921236434276200537897371193088
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^6 - x^4 - 2*x^3 - x^2 - 2*x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLRRRRC}$
$r_{\mathrm{on}}$:
3.930471319516693364871234452625797445704922732760487208392827514601283947617643474006912450624468022
comment: period $8$.
$\mathtt{RLLRRRRC}$
$c_{\mathrm{on}}$:
-1.896915538627477483553120354097771481340591228295339892839592795570451894574611351260690601539509518
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRRRC}$
$r_W$:
3.930472995732144840644021355896516382571134630536351525803027036728044993563183987910599569536004700
comment: period $8$.
$\mathtt{RLLRRRRC}$
$c_W$:
-1.896917994678832848351950512749427493238309249098198186635171971591942057636612517950412005378837068
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRRRC}$
$r_{\mathrm{pd}}$:
3.930478024327372919915052655578150994340366244402160516703063628417083535505703829374919645718297173
comment: period $8$.
$\mathtt{RLLRRRRC}$
$c_{\mathrm{pd}}$:
-1.896925362766415717758680134160733121641086630106107852153493493275274186188855816790910645543666536
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRRRC}$
$h_{\mathrm{top}}$:
0.5748640467968352377697804623968670130791744512058141617667781668766908560661850306087239767075093449
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^7 - x^6 - x^5 - x^4 + x^3 - x^2 + x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLRRLRC}$
$r_{\mathrm{on}}$:
3.944212196585010526408470603300072301166721867656622915230067447936626232702915413343117233559833465
comment: period $8$.
$\mathtt{RLLRRLRC}$
$c_{\mathrm{on}}$:
-1.917096364629983167405658942973954328676192150859132009516566131459369016123311436304666594500521168
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRLRC}$
$r_W$:
3.944213495732550988086109017328782885953259488294611311717779104444982533642044894925581646379964505
comment: period $8$.
$\mathtt{RLLRRLRC}$
$c_W$:
-1.917098277113422008833510932966779371236124465024502517689818098870808062416851814418466663315510217
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRLRC}$
$r_{\mathrm{pd}}$:
3.944217392716251503693647907641412771126567215884055599957619318355952086197007876146491813224455899
comment: period $8$.
$\mathtt{RLLRRLRC}$
$c_{\mathrm{pd}}$:
-1.917104013893220483238749258936981520671647711256654437709208055049456961361194387040826542444524455
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLRRLRC}$
$h_{\mathrm{top}}$:
0.5917189080612842698607150526771911119954496801779894835375029446718954303023597837038053235183539009
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^7 - x^6 - x^5 - x^4 + x^3 - x^2 - x + 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLRLLC}$
$r_{\mathrm{on}}$:
3.960768652406645256046371427380866877425475433389218910176759920822075912075442552844418007072453943
comment: period $8$; $r_{\mathrm{on}}$ is the first period-doubling point of $\mathtt{RLLC}$.
$\mathtt{RLLLRLLC}$
$c_{\mathrm{on}}$:
-1.941537753268465539334645081335137302112963652443093023862503797514245782358240271484061333910837329
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRLLC}$
$r_W$:
3.960933697547581466371142299135839172920423872079796785354269378905529636265236266311554757383705554
comment: period $8$.
$\mathtt{RLLLRLLC}$
$c_W$:
-1.941782090318198160140455136524972704898489796458099671661877339955525967487513658235591165702514575
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRLLC}$
$r_{\mathrm{pd}}$:
3.961098633486188820919718364687632275505865603796306323143193436067485131345044720979259133843380334
comment: period $8$.
$\mathtt{RLLLRLLC}$
$c_{\mathrm{pd}}$:
-1.942026279308443698811724818607598194944924281338460122297575548543814697630899686939604501633135288
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRLLC}$
$h_{\mathrm{top}}$:
0.6093778634360062315368033711683986954285392793128541477762892366225152051203195768397057073419585497
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^3 - x^2 - x - 1$; $h_{\mathrm{top}}=\log\lambda$; the entropy equals that of $\mathtt{RLLC}$.
$\mathtt{RLLLRLRC}$
$r_{\mathrm{on}}$:
3.973723762395857200730761432266542698680782241795843848875526856269484767386769043799732822418687560
comment: period $8$.
$\mathtt{RLLLRLRC}$
$c_{\mathrm{on}}$:
-1.960758253759443143125110248584684357683338793500136497251176178707866489242937422846941717351258127
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRLRC}$
$r_W$:
3.973724255674980337822883993721114490289705942652436601074755616440307058453306645045033896568229278
comment: period $8$.
$\mathtt{RLLLRLRC}$
$c_W$:
-1.960758987197428957479901627201746582819406657357862148103980091304574106073655390311192546255773545
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRLRC}$
$r_{\mathrm{pd}}$:
3.973725735423457405349991897874708592020131884557529470404052406035387518706979747221108971021760316
comment: period $8$.
$\mathtt{RLLLRLRC}$
$c_{\mathrm{pd}}$:
-1.960761187379945648277627837912193209561089051499896727039912535903910513898218292897772903136282491
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRLRC}$
$h_{\mathrm{top}}$:
0.6264426221026037992162732472562463584534961060224227767783113374562672064844081482539000123805726028
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^7 - x^6 - x^5 - x^4 - x^3 + x^2 + x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLRRRC}$
$r_{\mathrm{on}}$:
3.981408635979901517560970944275589368801122590531734351329518712228731634231697148277751478021283931
comment: period $8$.
$\mathtt{RLLLRRRC}$
$c_{\mathrm{on}}$:
-1.972199363673884229442993202153246443901891588911448865377835892255020002073736985685619600666981904
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRRRC}$
$r_W$:
3.981408954415261495395529301975011729042537023837915853842593414895880128174937863980003054458899338
comment: period $8$.
$\mathtt{RLLLRRRC}$
$c_W$:
-1.972199838366875699354940943628367727527556633676016311543483446546690420312525133230686983708021685
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRRRC}$
$r_{\mathrm{pd}}$:
3.981409909660411356397596171913212427274860784188635792022388925331314983361194804609546008047882536
comment: period $8$.
$\mathtt{RLLLRRRC}$
$c_{\mathrm{pd}}$:
-1.972201262355325551348997675275987706905933449974085928017966448025181916453064259872834062700147132
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRRRC}$
$h_{\mathrm{top}}$:
0.6391899326598130388240901324599952072591553666547079343960562150398365829981817920230864003961974078
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^7 - x^6 - x^5 - x^4 - x^3 + x^2 - x + 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLRRLC}$
$r_{\mathrm{on}}$:
3.987745285349480601383890365898478373959865927602208868204183274102352762106448594415709533212265539
comment: period $8$.
$\mathtt{RLLLRRLC}$
$c_{\mathrm{on}}$:
-1.981655472532012315772375630520963193419526119028740699823884279598876356424843480546761437928679654
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRRLC}$
$r_W$:
3.987745495370075483821404555582378116634167436580517047039344261876290664741215863536340880581136709
comment: period $8$.
$\mathtt{RLLLRRLC}$
$c_W$:
-1.981655786276044436082912388688275325921609985743449955431559578577448510223465800497120254259583688
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRRLC}$
$r_{\mathrm{pd}}$:
3.987746125417485357220439457873622419269484067589975022299741360683630535345330972837058944279911536
comment: period $8$.
$\mathtt{RLLLRRLC}$
$c_{\mathrm{pd}}$:
-1.981656727486799035433377886402877546588539423788459236747426580472726730993950798152094252176823795
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLRRLC}$
$h_{\mathrm{top}}$:
0.6517666834215227498951343928630231144891116678343853491206890411188226128861784636299763312446363640
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^6 - 2*x^5 + x^4 - 2*x^3 + x^2 - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLLRLC}$
$r_{\mathrm{on}}$:
3.992519399682352431486060579339922667741372238348303445577453540930719908932816422094346108727458752
comment: period $8$.
$\mathtt{RLLLLRLC}$
$c_{\mathrm{on}}$:
-1.988793089368806744454828551070683348459939994608405517498658240760089757329802350378329473707942136
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLRLC}$
$r_W$:
3.992519523284331897786690976275686086412434705786432527589167749683230334642511617609484898816711553
comment: period $8$.
$\mathtt{RLLLLRLC}$
$c_W$:
-1.988793274309471259839367775061205543127682526515804968022566929016540202791668452158141855395167388
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLRLC}$
$r_{\mathrm{pd}}$:
3.992519894066919479235621934021452162065156946924187555864997484869989701847519006871318738862282464
comment: period $8$.
$\mathtt{RLLLLRLC}$
$c_{\mathrm{pd}}$:
-1.988793829096571745455478829686267217671370544653881007784571223473312257526124406755894359412822792
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLRLC}$
$h_{\mathrm{top}}$:
0.6607909182862917105365364859711215906388029984208201560206927691524562768056083265969998626742249573
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^7 - x^6 - x^5 - x^4 - x^3 - x^2 + x + 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLLRRC}$
$r_{\mathrm{on}}$:
3.996219537052268091847193629658916159697825578251249095882869718271619053948697898626002473598046856
comment: period $8$.
$\mathtt{RLLLLRRC}$
$c_{\mathrm{on}}$:
-1.994332878553426931227818974259330289260069032919247854411117034960774413789517801301073694485491441
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLRRC}$
$r_W$:
3.996219595901164854911278514157144852312316466929969285351214084200151303506481380630949621508892713
comment: period $8$.
$\mathtt{RLLLLRRC}$
$c_W$:
-1.994332966715534904758276859241877571665259148564925261175230080782143422942012411283044531845570901
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLRRC}$
$r_{\mathrm{pd}}$:
3.996219772440470560913133895443558067395628147970171943516429440995508101289814403034164215501498347
comment: period $8$.
$\mathtt{RLLLLRRC}$
$c_{\mathrm{pd}}$:
-1.994333231190806297844434654763467395129400914575102166814895956757065266546038248023418317630536788
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLRRC}$
$h_{\mathrm{top}}$:
0.6713170136526259346020626331280442194228450020360553557542713414020113778740442359499853255561395849
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^5 - x^4 - 2*x^3 + x - 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLLLRC}$
$r_{\mathrm{on}}$:
3.998641615329287157332537506962064844319789177954475397981379422480568988528183406273015486875171084
comment: period $8$.
$\mathtt{RLLLLLRC}$
$c_{\mathrm{on}}$:
-1.997962884296159142908307188958435761662256416622457071765906476301717095440937665999064719775999703
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLLRC}$
$r_W$:
3.998641636206443602403492515258255085545110351024218029845030847683639128258331770147976601388927838
comment: period $8$.
$\mathtt{RLLLLLRC}$
$c_W$:
-1.997962915597714314837125904277217321267833449912631374749804326487668403724165048355291034539083718
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLLRC}$
$r_{\mathrm{pd}}$:
3.998641698835310571177688635533739721276620069612066301595808276764030087495753295664253851129699454
comment: period $8$.
$\mathtt{RLLLLLRC}$
$c_{\mathrm{pd}}$:
-1.997963009498479355931242992466397315111328187593140075182455807105596514868851953435127671276967973
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLLRC}$
$h_{\mathrm{top}}$:
0.6804766008728410576345498594714966317371684364472755687544788841692024862636447597868423347923600163
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^6 - 2*x^5 + x^4 - 2*x^3 + x^2 - 2*x + 1$; $h_{\mathrm{top}}=\log\lambda$.
$\mathtt{RLLLLLLC}$
$r_{\mathrm{on}}$:
3.999849359713914445493109243275757790484341865746318478140410284286476424899003930295164010405774898
comment: period $8$.
$\mathtt{RLLLLLLC}$
$c_{\mathrm{on}}$:
-1.999774045243995616224090560049811946191435106673003993694510185353213771596632693270589720218756095
comment: period $8$; the same window's $r_{\mathrm{on}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLLLC}$
$r_W$:
3.999849362013851070480287025220670591134590055617076966641977634592248041783594222732223795006028090
comment: period $8$.
$\mathtt{RLLLLLLC}$
$c_W$:
-1.999774048693727323471700996130915760674511960788419987606429651556462629287368141952989631187626142
comment: period $8$; the same window's $r_W$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLLLC}$
$r_{\mathrm{pd}}$:
3.999849368913379841119021607224808634940836343632346531758017858221390370719446070324724380060600499
comment: period $8$.
$\mathtt{RLLLLLLC}$
$c_{\mathrm{pd}}$:
-1.999774059042500825770984374146495497318926928283901010564309995633649261282137417184736886859922194
comment: period $8$; the same window's $r_{\mathrm{pd}}$ in the $z\mapsto z^2+c$ normalisation (1).
$\mathtt{RLLLLLLC}$
$h_{\mathrm{top}}$:
0.6891211854090296645856864820140183890319207822280265535590938317172931687130026903675465420860413724
comment: period $8$; if $\lambda$ is the Perron root then $\lambda$ satisfies $x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1$; $h_{\mathrm{top}}=\log\lambda$.
Definition
For $f_r(x)=rx(1-x)$ on $[0,1]$ [3], a period-$p$ window is an interval of $r$ where $f_r$ has an attracting orbit of exact period $p$. The parameter $W$ is the critical orbit\'s kneading word [5]. For each window the table holds where it opens, where it is superstable and where it first period-doubles, each in both the logistic parameter $r$ and the quadratic parameter $c$ of $z\mapsto z^2+c$ (1). It also holds the topological entropy $h_{\mathrm{top}}$ of $f_r$ on the window [4].
Parameters
$W$
—   kneading word
quantity
—   quantity
Formulas
(1)
$c=-\frac{r(r-2)}{4}$, so $r=4$ corresponds to $c=-2$ and $r=2$ corresponds to $c=0$.
(2)
$r_W$ satisfies $f_r^p(\frac12)=\frac12$, where $p$ is the period of the kneading word $W$.
(3)
The endpoints of a period-$p$ window satisfy $f_r^p(x)=x$ and $\prod_{j=0}^{p-1}r(1-2f_r^j(x))=\mu$, with $\mu=1$ for $r_{\mathrm{on}}$ and $\mu=-1$ for $r_{\mathrm{pd}}$.
(4)
$h_{\mathrm{top}}=\log\rho(A_W)$, where $A_W$ is the Markov transition matrix of the partition by the critical orbit of the superstable map and $\rho(A_W)$ is its Perron root [2].
(5)
For $W=\mathtt{RLC}$, $r_{\mathrm{on}}=1+2\sqrt2$, so $c_{\mathrm{on}}=-7/4$ exactly, and $h_{\mathrm{top}}=\log\varphi$ where $\varphi$ is the golden ratio $\varphi$.
Comments
(6)
The kneading word $W$ records the critical orbit: for a period-$p$ superstable orbit, its letters are $R$ or $L$ according as $f_r^j(1/2)$ lies to the right or left of $1/2$ for $1\leq j<p$, followed by $C$. The words are ordered by increasing $r_W$ within each period.
(7)
The onset $r_{\mathrm{on}}$ is where the period-$p$ orbit is born with multiplier $1$; the superstable parameter $r_W$ is where the orbit contains $1/2$; the first period-doubling parameter $r_{\mathrm{pd}}$ is where its multiplier is $-1$; and $h_{\mathrm{top}}$ is the topological entropy [4] of $f_r$ for $r$ in the window.
(8)
The omitted main period-doubling cascade windows have entropy $0$, and the ratios of successive parameter gaps in the cascade tend to the Feigenbaum constant $\delta$.
(9)
If $W$ is the doubled harmonic of a shorter word $V$, its onset is the first period-doubling point of $V$. Thus $\mathtt{RLLRLC}$ begins at the first period-doubling point of $\mathtt{RLC}$, and $\mathtt{RLLLRLLC}$ begins at the first period-doubling point of $\mathtt{RLLC}$. These harmonic windows have the same topological entropy as their parents.
(10)
The window $\mathtt{RLRRRC}$ is the period-$3$ window of the second iterate $f_r^2$; its topological entropy is half that of $\mathtt{RLC}$.
(11)
The two normalisations are the same window in different coordinates: $c=-r(r-2)/4$ by (1), so $r=4$ is $c=-2$ and $r=2$ is $c=0$. Both are rows rather than remarks, because a number written in a comment answers no search and cannot be cited. The entropy has no normalisation: it is a property of the map, and the same number whichever coordinate names the map.
(12)
OEIS gives the period-$3$ onset [8], the first period-doubling point of that window [9], and the first onsets of periods $5$, $6$ and $7$ [10] [11] [12]. The count of primitive kneading words by period is the sequence A000048 [7], in the Metropolis-Stein-Stein order [1].
Programs
(P1)
Sage
def f(r, x):
    return r*x*(1 - x)
def iterate(r, x, k):
    for _ in range(k):
        x = f(r, x)
    return x

find_root(lambda r: iterate(r, 1/2, 3) - 1/2,
          3.83, 3.84)                         # superstable RLC
find_root(lambda r: r^6 - 6*r^5 + 4*r^4 + 24*r^3
          - 14*r^2 - 36*r - 81, 3.84, 3.85)   # first doubling of RLC
log((1 + sqrt(5))/2)                           # entropy of RLC
References
[1]
Nicholas Metropolis, M. L. Stein and P. R. Stein, "On finite limit sets for transformations on the unit interval", Journal of Combinatorial Theory, Series A 15 (1973), 25-44. (doi) (MR)
[2]
John Milnor and William Thurston, "On iterated maps of the interval", Dynamical Systems, Lecture Notes in Mathematics 1342, Springer, 1988, 465-563. (doi) (MR)
Links
Similar tables
Feigenbaum constants —   the first period-doubling endpoint of each window begins a local period-doubling cascade with the same Feigenbaum ratio $\delta$
Golden ratio —   the entropy of the $\mathtt{RLC}$ window is $\log\varphi$
Entropy constants of lattice models —   the entropy of the $\mathtt{RLC}$ window, $\log\varphi$, is the hard-core line entropy; both describe the golden-mean shift
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds every primitive superstable kneading word with period at most $8$, except the main period-doubling cascade words, with seven quantities for each word)
How they were obtained:

The superstable rows were computed by Newton iteration on (2). The onset and first period-doubling rows were computed by Newton continuation in the multiplier $\mu$ in (3); for the two harmonic windows $\mathtt{RLLRLC}$ and $\mathtt{RLLLRLLC}$, the onset is the parent's first period-doubling point because the multiplier-$1$ continuation is singular there. The entropy rows were computed as $\log\rho(A_W)$ from the integer Markov transition matrix in (4). Every row was recomputed with $760$ and $980$ guard bits beyond the $100$ written digits, and the written digits agreed. An independent check built the superstable polynomials $f_r^p(1/2)-1/2$ exactly, found the real roots of exact period $p$ for $3\leq p\leq8$, recovered the same $34$ retained kneading words, and checked the counts $1,2,3,5,9,16$ against OEIS A000048 [7]. The period-$3$ onset was checked against $1+2\sqrt2$ [8]; the period-$3$ first period-doubling point was checked against its sextic equation [9]; the first period-$5$, $6$ and $7$ onsets were checked against the OEIS digits named in the links; and the entropy controls $h_{\mathrm{top}}(\mathtt{RLC})=\log\varphi$, $h_{\mathrm{top}}(\mathtt{RLRRRC})=\frac12\log\varphi$, and the two harmonic entropy equalities were checked.

more

The $c$ rows are $-r(r-2)/4$ applied to the $r$ row beside them, in the same arithmetic, so they are no weaker and no stronger than it. One of them is exact: $\mathtt{RLC}$ opens at $r=1+2\sqrt2$, so $c_{\mathrm{on}}=-(1+2\sqrt2)(2\sqrt2-1)/4=-(8-1)/4=-7/4$ and the surd cancels; it is written as the rational and checked against the solve.



API Python package Skill for AI Privacy Legal notice Source
Data licensed CC BY-SA 4.0. No tracking, no analytics, essential cookies only.