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Lecture Notes in Mathematics 2214 Saeed Zakeri Rotation Sets and Complex Dynamics Lecture Notes in Mathematics 2214 Editors-in-Chief: Jean-MichelMorel,Cachan BernardTeissier,Paris AdvisoryBoard: MichelBrion,Grenoble CamilloDeLellis,Zurich AlessioFigalli,Zurich DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Athens GáborLugosi,Barcelona MarkPodolskij,Aarhus SylviaSerfaty,NewYork AnnaWienhard,Heidelberg Moreinformationaboutthisseriesathttp://www.springer.com/series/304 Saeed Zakeri Rotation Sets and Complex Dynamics 123 SaeedZakeri DepartmentofMathematics QueensCollegeofCUNY Queens,NY USA DepartmentofMathematics GraduateCenterofCUNY NewYork,NY USA ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics ISBN978-3-319-78809-8 ISBN978-3-319-78810-4 (eBook) https://doi.org/10.1007/978-3-319-78810-4 LibraryofCongressControlNumber:2018939069 MathematicsSubjectClassification(2010):37E10,37E15,37E45,37F10 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAGpart ofSpringerNature. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface For an integer d ≥ 2, let m : R/Z → R/Z denote the multiplication by d map d ofthe circledefinedbym (t) = dt (mod Z). A rotationset form isa compact d d subset of R/Z on which m acts in an order-preservingfashion and thereforehas d a well-defined rotation number. Rotation sets for the doubling map m seem to 2 have first appeared under the disguise of Sturmian sequences in a 1940 paper of MorseandHedlundonsymbolicdynamics[17] (theequivalencewiththerotation setconditionwaslater shownbyGambaudoetal. [10] andVeerman[28]).Fertile ground for their comeback was provided half a century later by the resurgence of the field of holomorphic dynamics. For example, in the early 1990s Goldberg [11] and Goldberg and Milnor [12] studied rational rotation sets in their work on fixed point portraits of complex polynomials. The main result of [11] was later extendedbyGoldbergandTressertoirrationalrotationsets[13].Aroundthesame time,BullettandSentenacinvestigatedrotationsetsforthedoublingmapandtheir connectionwith theDouady–HubbardtheoryoftheMandelbrotset[7] (see Fig.1 for an illustration of this link). Aspects of this work were generalized to arbitrary degreesadecadelaterbyBlokhetal.whoinparticulargaverecipesforconstructing arotationsetformd+1fromoneformd andviceversa[2].Morerecently,Bonifant, Buff,andMilnorusedrotationsetsforthetriplingmapm intheirworkonantipode- 3 preservingcubicrationalmaps[4].Inanentirelydifferentcontext,rationalrotation sets appear in McMullen’s study of the space of proper holomorphicmaps of the unit disk [19]; they play a role analogousto simple closed geodesics on compact hyperbolicsurfaces. Thismonographpresentsthefirstsystematictreatmentofthetheoryofrotation sets for m in both rational and irrational cases. Our approach, partially inspired d bythe ideasin [4], hasa rathergeometricflavor andyieldsseveralnew resultson the structure of rotation sets, their gap dynamics, maximal and minimal rotation sets, rigidity, and continuous dependence on parameters. This “abstract” part is supplementedwith a “concrete”partwhich explainshow rotationsets arise in the dynamicalplaneofcomplexpolynomialmapsandhowsuitableparameterspacesof suchpolynomialsprovideacompletecatalogofallrotationsetsofagivendegree. v vi Preface 10/31 9/31 5/31 ω 18/31 20/31 ω+1/2 θ=2/5 θ =(√5−1)/2 10/31 9/31 5/31 ω 0 0 18/31 ω+1/2 20/31 Fig.1 Foreach0 ≤ θ < 1thedoublingmapt (cid:5)→ 2t (mod Z)hasauniqueminimalinvariant set Xθ ⊂ R/Z of rotation number θ which is a period orbit if θ is rational and a Cantor set otherwise.Topleft:Thecaseθ = 2/5w√hereXθ isthe5-cycle 351 (cid:5)→ 1301 (cid:5)→ 2301 (cid:5)→ 391 (cid:5)→ 1381. Topright:Thegoldenmeancaseθ = 52−1 wheretheCantorsetXθ istheclosureoftheorbit ofω≈0.35490172....AccordingtoDouadyandHubbard,the“rotationset”Xθ isrelatedtothe externalraysofthecorrespondingquadraticmapz(cid:5)→e2πiθz+z2(showninthebottomrow)as wellastheparameterraysthatlandontheboundaryofthemaincardioidoftheMandelbrotset. SeeSect.5.3fordetails Hereisanoutlineofthematerialpresentedinthismonograph: Chapter1providesbackgroundmaterialonthedynamicsofdegree1monotone maps of the circle. Given such a map g : R/Z → R/Z, its Poincaré rotation number ρ(g) is constructed using a Dedekind cut approach that quickly leads to basic properties of the rotation number and how it essentially determines the asymptotic behavior of the orbits of g. These orbits converge to a cycle if ρ(g) is rational and to a unique minimal Cantor set if ρ(g) is irrational. A key tool in understandingthisdichotomyisthesemiconjugacybetweengandtherigidrotation r :t (cid:5)→t+θ (mod Z)bytheangleθ =ρ(g).Thissemiconjugacyisalsoutilized θ Preface vii in studying the existence and uniqueness of invariant probability measures for g: Ifρ(g)isrational,everysuchmeasureisaconvexcombinationofDiracmeasures supportedonthecyclesofg,whileifρ(g)isirrational,thereisauniqueinvariant measuresupportedontheminimalCantorsetofg. Chapter 2 introduces rotation sets for the map m and develops their basic d properties. A rotation set for m is a non-empty compact set X ⊂ R/Z, with d m (X) = X, such that the restriction m | extendsto a degree 1 monotonemap d d X ofthecircle.TherotationnumberofX,denotedbyρ(X),isdefinedastherotation number of any such extension. We refer to X as a rational or irrational rotation set according as ρ(X) is rational or irrational. Understanding X is facilitated by studying the dynamics of the complementary intervals of X called its gaps. A gap I is labeled minor or major according as m | : I → m (I) is or is not a d I d homeomorphism,andthemultiplicityofI isthenumberoftimesthecoveringmap m wraps I around the circle. Counting multiplicities, X has d − 1 major gaps, d a statement reminiscent of the fact that a degree d polynomial has d − 1 critical points.Majorgapscompletelydeterminearotationsetandthepatternofhowthey aremappedaroundcanberecordedinacombinatorialobjectcalledthegapgraph. Next, we study maximal and minimal rotation sets. Maximal rotation sets for m are characterized as having d − 1 distinct major gaps of length 1/d. A d rationalrotationsetmaywellbecontainedininfinitelymanymaximalrotationsets. By contrast,we show that an irrationalrotationset form is containedin at most d (d−1)!maximalrotationsets.Minimalrotationsetsarecyclesintherationalcase andCantorsetsintheirrationalcase.Weprovethatarationalrotationsetcontains atmostasmanyminimalrotationsetsasthenumberofitsdistinctmajorgaps.As a special case, we recoverGoldberg’sresult in [11] accordingto which a rational rotationsetform containsatmostd−1cycles.Ontheotherhand,everyirrational d rotationsetiseasilyshowntocontainauniqueminimalrotationset. Chapter3offersamorein-depthstudyofminimalrotationsetsbypresentinga unifiedtreatmentof thedeploymenttheoremofGoldbergandTresser. SupposeX is a minimal rotation set for m with the rotation number ρ(X) = θ (cid:9)= 0. Then d X is a q-cycle(i.e.,a cycle of lengthq)if θ = p/q in lowesttermsanda Cantor set if θ is irrational. The natural measure on X is the unique m -invariantBorel d probability measure μ supported on X. The canonical semiconjugacy associated withXisadegree1monotonemapϕ :R/Z→R/Z,whoseplateausareprecisely the gaps of X, which satisfies ϕ ◦m = r ◦ϕ on X. It is related to the natural d θ measurebyϕ(t) = μ[0,t](mod Z).Thecoveringmapm hasd −1fixedpoints d u = i/(d − 1) (mod Z). The deployment vector of X is the probability vector i δ(X) = (δ1,...,δd−1) where δi = μ[ui−1,ui). Note that qδ(X) ∈ Zd−1 if θ is rationaloftheformp/q. Thedeploymenttheoremassertsthatgivenanyθ andanyprobabilityvectorδ ∈ Rd−1 thatsatisfies qδ ∈ Zd−1 if θ = p/q, thereexistsa uniqueminimalrotation set X = X for m with ρ(X) = θ and δ(X) = δ. The rational case of this θ,δ d theoremthatappearsin[11]anditsirrationalcaseprovedin[13]aretreatedusing very different arguments. By contrast, we provide a proof that reveals the nearly viii Preface identical nature of the two cases. The key tool in our unified treatmentis the gap measure d(cid:2)−1(cid:2)∞ ν = d−(k+1)1σi−kθ, i=1k=0 where σ = δ + ···+ δ and 1 denotes the unit mass at x. This is an atomic i 1 i x measure supported on the union of at most d −1 backward orbits of the rotation r .Thegeneralideaisthatthegapmeasurecanbeusedtoconstructthe“inverse” θ of the canonical semiconjugacyof X and therefore X itself. This measure makes a briefappearancein an appendixof [13], butits realpoweris notnearly utilized there. In addition to its theoretical role, the gap measure turns out to be a highly effectivecomputationalgadget. Chapter3alsoincludesafairlydetaileddiscussionoffiniterotationsets,namely, unionsofcyclesthathaveawell-definedrotationnumber.LetC (p/q)denotethe d collection of all q-cycles under m with rotation number p/q. According to the d deploymenttheorem,Cd(p/q)canbeidentifiedwith(cid:3)afinitesubsetofthe(cid:4)simplex(cid:5) Δd−2 = {(x1,...,xd−1) ∈ Rd−1 : xi ≥ 0 and di=−11xi = 1} with q+qd−2 elements. A collection of cycles in C (p/q) are compatible if their union forms d a rotation set. In [19], McMullen proposes that C (p/q) can be identified with d the vertices of a simplicial subdivision Δd−2 of Δd−2, where each collection of q compatible cycles corresponds to the vertices of a simplex in Δd−2. We provide q a justification for this geometric model; in particular, for each x ∈ Δd−2 our proof gives an explicit algorithm for finding a simplex in Δd−2 that contains x. q The subdivisionΔd−2 is differentfrom(andin a sense simpler than)the standard q barycentricsubdivisionandcouldperhapsbeofindependentinterestinapplications outsidedynamics. In Chap.4, we give sample applications of the results of Chaps.2 and 3, especially the deployment theorem. For example, we show that every admissible graph without loops can be realized as the gap graph of an irrational rotation set. We also study the dependence of the minimal rotation set X on the parameter θ,δ (θ,δ). We prove that the map (θ,δ) (cid:5)→ X is lower semicontinuous in the θ,δ Hausdorff topology, and it is continuous at some parameter (θ ,δ ) if and only 0 0 if X is exact in the sense that it is both minimal and maximal. We provide a θ0,δ0 characterizationof exactnesswhich showsthat the set of such parametershasfull measurein(R/Z)×Δd−2. Asanotherapplication,weusethegapmeasuretocomputetheleadingangleω ofX =X ,thatis,thesmallestanglewhenXisidentifiedwithasubsetof(0,1): θ,δ d(cid:2)−1 (cid:2) 1 N 1 N ω= ν(0,θ]+ 0 = d−(k+1)+ 0 . d−1 d −1 d−1 d−1 i=10<σi−kθ≤θ Preface ix Here,N ≥0isthenumberofindicesi withσ =0.Theformulagivesanexplicit 0 i algorithm for computing the base-d expansion of the angle (d − 1)ω, which has an itinerary interpretation in the context of polynomial dynamics. We exploit the leading angle formula in the low-degree cases d = 2 and d = 3 to carry out a detailed analysis of the structure of minimal rotation sets under the doubling and triplingmaps. Chapter5exploresthelinkbetweenrotationsetsandcomplexpolynomialmaps. After a brief review of the basic definitions in polynomial dynamics, we explain howanindifferentfixedpointofapolynomialofdegreed determinesarotationset underm . Moreprecisely,theanglesofthedynamicraysthatlandona parabolic d point or on the boundary of a “good” Siegel disk define a rotation set X with ρ(X)=θ, where e2πiθ is the multiplier of the parabolicpointor the center of the Siegeldisk.Intheparaboliccase,thisstatementiswellknownandgoesbacktothe workofGoldbergandMilnor[12].TheSiegelcase,whilesimilarinspirit,istrickier becauseofthepossibilityofraysaccumulatingbutnotlandingontheboundary.The “good”Siegel disk assumption refersto a limb decompositionhypothesis,similar toMilnor’sin[22],whichallowsustoprovethe requiredlandingstatements(this hypothesisisweakerthanlocalconnectivityoftheJuliasetandpresumablyholds forLebesguealmosteveryθ).Thedeploymentvectorδ(X)canberecoveredfrom theinternalanglesofthemarkedrootsontheboundaryoftheSiegeldisk,asseen fromitscenter. Thesegeneralremarksareillustratedingreaterdetailintwolow-degreefamilies ofpolynomialmaps.AccordingtoDouadyandHubbard,thecombinatorialstructure oftheMandelbrotset(specifically,theboundaryofthemaincardioidandthelimbs growing from it) catalogs all rotation sets under the doubling map m (see [9] 2 and [20]). We give a brief account of this in a section on the quadratic family, setting the stage for the simplest higher degree example, namely, the family of cubicpolynomialswith anindifferentfixedpointof a givenrotationnumber.This one-dimensional slice was studied in [30] in the irrational case and has been the subjectofinvestigationsbyothers(seeforexample[6]).Thereareindeedintriguing connectionsbetweenrotationsetsunderthetriplingmapm andthiscubicfamily. 3 Fix0 < θ < 1andconsiderthespaceofmoniccubicpolynomialswithafixed point of multiplier e2πiθ at the origin. Each such cubic has the form f : z (cid:5)→ a e2πiθz+az2+z3 forsomea ∈ C,wherefa andf−a areaffinelyconjugateunder theinvolutionz(cid:5)→−z.Theconnectednesslocus M (θ)={a ∈C:TheJuliasetJ(f )isconnected} 3 a iscompact,connected,andfull(compareFigs.5.8and5.10).OutsideM (θ)exactly 3 one critical pointof f escapes to ∞ and the Böttcher coordinateof the escaping a co-criticalpointgivesaconformalisomorphismC(cid:2)M (θ) → C(cid:2)Dwhichcan 3 beusedtodefinetheparameterraysofM (θ). 3 When θ is rational of the form p/q in lowest terms, the set X of angles of a dynamic rays that land at the parabolic point 0 is a rotation set under tripling with ρ(X ) = p/q. There are 2q +1 possibilities for X parametrized by their a a