They say the mset is fully connected, so shouldn't the calculated areas touch (on a single point)?

The connectedness of the Mset proves that there is a path - between all the marked components - that never leaves the Mset (including its boundary). But as the images are point-sampled, there is always distance between two numbers for two adjacent pixels whose escape behaviour is not depicted in the image. So there could be a chain of components, just pretending that the interesting components touch.

For now, I would need the equations for the boundaries of those components to verify that they intersect (similar to

forum link) - however, those are, afaik, not computed (computable).

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The cosine of \( \angle (c_{14}\rightarrow c_7,c_{14}\rightarrow c_{28}) \) was calculated via interval arithmetics and outward rounding to be (a lot more digits were identical but by far not necessary):

`[-0.997681155858468473453590591576224796926146179..-0.997681155858468473453590591576224796926146178]`

which does neither contain -1 nor 1 and hence the angle is different from

**180 (and 0) degree** and the three components' center do not form a straight line.

It would be interesting to see, where the next period-doubled bulbs locate. How does the angle between two consecutive components change? Maybe some angle bends in the opposite direction, so that the chain of centers does not deviate too much from a line?

I think they cannot always bend to the same side as then a spiral would form and the Mset ended there..

Algorithmic details- a small pixel neighbourhood around the numerically found hyperbolic center approximation was computed and the rectangle then subjected to IA subdivision root finding. Initial regions for subdivision were pairwise disjoint.
- a tailor-made high precision fix-point number type was used at 5400 integer decimals and 432 fractionals.
- the center for period p was analyzed with the polynomial describing the hyperbolic centers of period p and all its divisiors. Once a root region B was found at the expected period, the polynomials for the divisors (and their divisors) were tested to verfiy that B does not contain a hyperbolic center of divisor period.
- as a confirmation, for periods 7 and 14, the cell-mapping TSA was used to verify that the final root region (outward rounded to fit double precision) only harbours Mset numbers that exhibit a period p cycle. For period 28, at my current hard- and software limitation at L20, no cycle could be detected.
- as a negative control, the center regions were used at only 5 fractional digits and provided a cosine straddling -1.

The root regions [r0..r1]+i*[i0..i1] below reliably enclose the respective hyperbolic centers at an accuracy of better than 10^-214.

`// root region c_7:`

r0 =0.432376192641994507824669648086921373887850639876994036204241251002834166302290895459969880303985995847338524802499444595200139126565289328917522446637247277753980718637680733065714274518276324279285081052783625783121513424012188391558069593824380745958530100454254238044374424786558161055111526716393202087428806028412316685703111009341084146259126312336041409904148286853260661785467832218239446642892087439714838594933819516297044

r1 =0.432376192641994507824669648086921373887850639876994036204241251002834166302290895459969880303985995847338524802499444595200139126565289328917522446637247277753980718637680733065714274518276324279285081052783625783121513424012188391558069648428917485674576039785791878850040668940564750026196306054719933688170910324060790346329317130140865968857126358491381615883279107259383922862591976490966012725662050413597983312182379224654431

i0=-0.226759904435348618697876559971698972120232191460389969044495193517667854349636330672103012270364858819519874060651583702193268713427265050756625137903812712764755318330898876406456506084088377450876489339901304121596135307743420618839822328177985217816158378759852162298045714727826739096886913258157027118769496626929545737508142300141583121198700990476858360963121728470907468531139252016557772603451162537976660814331862144136271

i1=-0.226759904435348618697876559971698972120232191460389969044495193517667854349636330672103012270364858819519874060651583702193268713427265050756625137903812712764755318330898876406456506084088377450876489339901304121596135307743420618839822258566458208371156535664292813797169939879876703259529201843219683736636987920502527659463190369717022309058064190379368206284531554011808440523360133916508583258631679236555497017441223702361499

// root region c_14:

r0 =0.432568811799922736600099472465418444280795099414752108944054334585526076638071723523287688129803072713971685785956316187407730111564636458980274281505126053906187688758386000412447372294514132443493865457899983686386050916949753506107593999604298200655101640658366651637204928066312141756640378889717076471108698548286598813495831024576945358943372800169544662714158665148836149220156531017677638232777499872570573361622335097447913

r1 =0.432568811799922736600099472465418444280795099414752108944054334585526076638071723523287688129803072713971685785956316187407730111564636458980274281505126053906187688758386000412447372294514132443493865457899983686386050916949753506107593999604298200655101640658366651637204928066312141756640378889717076471108698548286598813495831024576945358943372800779146711133940622122276961669135766270437742741644311805768657847986576171336495

i0=-0.228734414339756139285265418096148959018123393842058177114661793624735462497185051266270383039206071792914416488180703781037657604227588208595148610615155847887721546278565051713880906906104420106955493267214789196215659070055734556151016312766072935438630867350108119526500824409548776941297019146391234284903304515024231182037023498097923423484740988431831757871966960814682954475921467796447502217656699540255738662017375937785323

i1=-0.228734414339756139285265418096148959018123393842058177114661793624735462497185051266270383039206071792914416488180703781037657604227588208595148610615155847887721546278565051713880906906104420106955493267214789196215659070055734556151016312766072935438630867350108119526500824409548776941297019146391234284903304515024231182037023498097923423484740987685550434107796026172344084396557335131124140011627804186747904087068974868526327

// root region c_28:

r0 =0.432669818784130669539677962288015718057584368534966008335078278147638921849466453209300644500416434899611715441943009051869858029361248696871454534866877623543225372268656684228338700662282974572960913707473852413978391956252438172554546064149824172457766594454427356267484815805755488837322130854030605498865141101363580658296114102512043236813490826220061135922666561256473986569865906720610501785676014870798219943630611090734779

r1 =0.432669818784130669539677962288015718057584368534966008335078278147638921849466453209300644500416434899611715441943009051869858029361248696871454534866877623543225372268656684228338700662282974572960913707473852413978473139337829328627509075418224600380736679482038998934418198544590908053059736038792758161280639154211983975036318849897695717533930373772224977764025919042942439457255636396029345632867674903706921544266519580811572

i0=-0.229339671447065501400885319925200893099553632310193579062445535224188960174462845810881314954535721712263062955421632200010042794928160912087383920767579040479821377825587723064616206222728195265212927630534782657281923332501488170233368706683818113112125843598537213654856258848813460811313818992217552231606073336971381071066045756676336224608296534439178712742418054472400135114703238164679059867458697420625368367611296280988619

i1=-0.229339671447065501400885319925200893099553632310193579062445535224188960174462845810881314954535721712263062955421632200010042794928160912087383920767579040479821377825587723064616206222728195265212927630534782657281824538063636007533208913160748769660019022246491827545586227530730990738006378188114698284822608246592829814630106994369699471619460547041174953475469170650393206335951435865862177808936735285455392087009645858240357