Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random 9 Julia sets which are Jordan curves but not quasicircles 0 ∗ 0 2 Hiroki Sumi c Department of Mathematics, Graduate School of Science e D Osaka University 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan 9 E-mail: [email protected] ] http://www.math.sci.osaka-u.ac.jp/ sumi/ S ∼ D December 8, 2009 . h t a m Abstract [ Weinvestigatethedynamicsofpolynomial semigroups(semigroups generated byafamily 4 of polynomial maps on the Riemann sphere Cˆ) and the random dynamics of polynomials on v the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) 6 dynamics, we give a classification of polynomial semigroups G such that G is generated by a 3 5 compact family Γ, the planar postcritical set of G is bounded, and G is (semi-) hyperbolic. 4 In one of the classes, we have that for almost every sequence γ ∈ΓN, the Julia set Jγ of γ is 1. a Jordan curve but not a quasicircle, the unbounded component of Cˆ \Jγ is a John domain, 1 andtheboundedcomponentofC\Jγ isnotaJohndomain. Notethatthisphenomenondoes not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics 8 0 of polynomial semigroups G such that the planar postcritical set of G is bounded and the : Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics v of polynomials that do not occur in the usual dynamics of polynomials are systematically i X investigated. r a 1 Introduction This is the third paper in which the dynamics of semigroups of polynomial maps with bounded planar postcritical set in C are investigated. This paper is self-contained and the proofs of the results of this paper are independent from the results in [37, 38]. The theoryofcomplex dynamicalsystems,whichhas its origininthe importantworkofFatou andJuliainthe1910s,hasbeeninvestigatedbymanypeopleanddiscussedindepth. Inparticular, since D. Sullivan showedthe famous “no wandering domain theorem” using Teichmu¨ller theory in the 1980s, this subject has attracted many researchers from a wide area. For a general reference on complex dynamical systems, see Milnor’s textbook [16]. There are several areas in which we deal with generalized notions of classical iteration theory of rational functions. One of them is the theory of dynamics of rational semigroups (semigroups generated by a family of holomorphic maps on the Riemann sphere Cˆ), and another one is the theory of random dynamics of holomorphic maps on the Riemann sphere. ∗To appear in Ergodic Theory Dynam. Systems. 2000 Mathematics Subject Classification. 37F10, 30D05. Keywords: Complexdynamics,polynomialsemigroup,rationalsemigroup,Randomcomplexdynamics,Juliaset. 1 In this paper, we will discuss these subjects. A rational semigroupis a semigroupgenerated by a family of non-constant rational maps on Cˆ, where Cˆ denotes the Riemann sphere, with the semigroup operation being functional composition ([12]). A polynomial semigroup is a semigroup generated by a family of non-constant polynomial maps. Research on the dynamics of rational semigroups was initiated by A. Hinkkanen and G. J. Martin ([12, 13]), who were interestedintheroleofthedynamicsofpolynomialsemigroupswhilestudyingvariousone-complex- dimensional moduli spaces for discrete groups, and by F. Ren’s group([45, 11]), who studied such semigroupsfromtheperspectiveofrandomdynamicalsystems. Moreover,theresearchonrational semigroups is relatedto that on“iteratedfunction systems”in fractalgeometry. In fact, the Julia set of a rational semigroup generated by a compact family has “ backward self-similarity” (cf. Lemma 3.1-2). For other research on rational semigroups, see [20, 21, 22, 44, 23, 24, 42, 41, 43], and [27]–[39]. The research on the dynamics of rational semigroups is also directly related to that on the random dynamics of holomorphic maps. The first study in this direction was by Fornaess and Sibony ([9]), and much researchhas followed. (See [1, 3, 4, 2, 10, 34, 39].) We remark that the complex dynamical systems can be used to describe some mathematical models. For example, the behavior of the population of a certain species can be described as the dynamicalsystemofapolynomialf(z)=az(1 z)suchthatf preservesthe unit intervalandthe − postcritical set in the plane is bounded (cf. [8]). It should also be remarkedthat according to the change of the natural environment, some species have several strategies to survive in the nature. Fromthispointofview,itisveryimportanttoconsidertherandomdynamicsofsuchpolynomials (see also Example 1.4). For the random dynamics of polynomials on the unit interval, see [26]. We shall give some definitions for the dynamics of rational semigroups. Definition 1.1 ([12, 11]). Let G be a rational semigroup. We set F(G)= z Cˆ G is normal in a neighborhood of z , and J(G)=Cˆ F(G). { ∈ | } \ F(G) is called the Fatou set of G and J(G) is called the Julia set of G. We let h ,h ,... 1 2 h i denote the rationalsemigroupgeneratedby the family h .Moregenerally,fora family Γ ofnon- i { } constant rational maps, we denote by Γ the rational semigroup generated by Γ. The Julia set of h i the semigroup generated by a single map g is denoted by J(g). Similarly, we set F(g):=F( g ). h i Definition 1.2. 1. For each rational map g : Cˆ Cˆ, we set CV(g) := all critical values of g : Cˆ Cˆ . → { → } Moreover,for each polynomial map g :Cˆ Cˆ, we set CV (g):=CV(g) . ∗ → \{∞} 2. Let G be a rational semigroup. We set P(G):= CV(g) ( Cˆ). ⊂ g[G ∈ This is calledthe postcritical setofG.Furthermore,fora polynomialsemigroupG,we set P (G) := P(G) . This is called the planar postcritical set (or finite postcritical ∗ \{∞} set) of G. We say that a polynomial semigroup G is postcritically bounded if P (G) is ∗ bounded in C. Remark 1.3. Let G be a rational semigroup generated by a family Λ of rational maps. Then, we have that P(G)= g( CV(h)), where Id denotes the identity map on Cˆ. Thus g G Id h Λ ∈ ∪{ } ∈ g(P(G)) P(G) for eSach g G. FrSom this formula, one can figure out how the set P(G) (resp. ⊂ ∈ P (G)) spreads in Cˆ (resp. C). In fact, in Section 5, using the above formula, we present a ∗ waytoconstructexamplesofpostcriticallyboundedpolynomialsemigroups(with someadditional properties). Moreover, from the above formula, one may, in the finitely generated case, use a computer to see if a polynomial semigroup G is postcritically bounded much in the same way as one verifies the boundedness of the critical orbit for the maps f (z)=z2+c. c 2 Example 1.4. LetΛ:= h(z)=cza(1 z)b a,b N, c>0, c( a )a( b )b 1 andletGbethe { − | ∈ a+b a+b ≤ } polynomial semigroup generated by Λ. Since for each h Λ, h([0,1]) [0,1] and CV (h) [0,1], ∗ ∈ ⊂ ⊂ it follows that each subsemigroup H of G is postcritically bounded. Remark 1.5. It is well-known that for a polynomial g with deg(g) 2, P ( g ) is bounded in C ∗ ≥ h i if and only if J(g) is connected ([16, Theorem 9.5]). AsmentionedinRemark1.5,theplanarpostcriticalsetisonepieceofimportantinformationre- gardingthedynamicsofpolynomials. Concerningthetheoryofiterationofquadraticpolynomials, we have been investigating the famous “Mandelbrot set”. When investigating the dynamics of polynomial semigroups, it is natural for us to discuss the relationship between the planar postcritical set and the figure of the Julia set. The first question in this regard is: Question 1.6. Let G be a polynomial semigroup such that each element g G is of degree two or more. Is J(G) necessarily connected when P (G) is bounded in C? ∈ ∗ The answer is NO. Example 1.7 ([44]). Let G = z3,z2 . Then P (G) = 0 (which is bounded in C) and J(G) h 4 i ∗ { } is disconnected (J(G) is a Cantor set of round circles). Furthermore, according to [32, Theorem 2.4.1], it can be shown that a small perturbation H of G still satisfies that P (H) is bounded in ∗ C and that J(H) is disconnected. (J(H) is a Cantor set of quasi-circles with uniform dilatation.) Question 1.8. What happens if P (G) is bounded in C and J(G) is disconnected? ∗ Problem 1.9. Classify postcritically bounded polynomial semigroups. Definition 1.10. Let be the set of all polynomial semigroups G with the following properties: G each element of G is of degree two or more, and • P (G) is bounded in C, i.e., G is postcritically bounded. ∗ • Furthermore,weset = G J(G) is connected and = G J(G) is disconnected . con dis G { ∈G | } G { ∈G | } We also investigate the dynamics of hyperbolic or semi-hyperbolic polynomial semigroups. Definition 1.11. Let G be a rational semigroup. 1. We say that G is hyperbolic if P(G) F(G). ⊂ 2. We say that G is semi-hyperbolic if there exists a number δ > 0 and a number N N ∈ such that, for each y J(G) and each g G, we have deg(g : V B(y,δ)) N for each ∈ ∈ → ≤ connected component V of g 1(B(y,δ)), where B(y,δ) denotes the ball of radius δ with − centery with respectto the sphericaldistance,anddeg(g : )denotesthe degreeoffinite ·→· branched covering. (For the backgroundof semi-hyperbolicity, see [27] and [30].) Remark 1.12. There are many nice properties of hyperbolic or semi-hyperbolic rational semi- groups. For example, for a finitely generated semi-hyperbolic rational semigroup G , there exists an attractor in the Fatou set ([27, 30]), and the Hausdorff dimension dim (J(G)) of the Julia H set is less than or equal to the critical exponent s(G) of the Poincar´e series of G ([27]). If we assume further the “open set condition”, then dim (J(G)) =s(G) ([33, 43]). Moreover, if G H is generated by a compact set Γ and if G is semi-hyperbolic, then for each sequence γ ΓN,∈thGe ∈ basin of infinity for γ is a John domain and the Julia set of γ is connected and locally connected ([30]). This fact will be used in the proofs of the main results of this paper. 3 In this paper, we classify the semi-hyperbolic, postcritically bounded, polynomial semigroups generated by a compact family Γ of polynomials. We show that such a semigroup G satisfies either (I) every fiberwise Julia set is a quasicircle with uniform distortion, or (II) for almost every sequence γ ΓN, the Julia set J is a Jordan curve but not a quasicircle, the basin of infinity A γ γ ∈ is a John domain, and the bounded component U of the Fatou set is not a John domain, or (III) γ for every α,β ΓN, the intersection of the Julia sets J and J is not empty, and J(G) is arcwise α β ∈ connected(cf. Theorem2.19). Furthermore,wealsoclassifythehyperbolic,postcriticallybounded, polynomial semigroups generated by a compact family Γ of polynomials. We show that such a N semigroup G satisfies either (I) above, or (II) above, or (III)’: for every α,β Γ , the intersection oftheJuliasetsJ andJ isnotempty,J(G)isarcwiseconnected,andforev∈erysequenceγ ΓN, α β ∈ thereexistinfinitelymanyboundedcomponentsofF (cf. Theorem2.21). Wegivesomeexamples γ of situation (II) above (cf. Example 2.22, figure 1, Example 2.23, and Section 5). Note that situation (II) above is a special phenomenon of random dynamics of polynomials that does not occur in the usual dynamics of polynomials. The key to investigating the dynamics of postcritically bounded polynomial semigroups is the density of repelling fixed points in the Julia set (cf. Theorem 3.2), which can be shown by an application of the Ahlfors five island theorem, and the lower semi-continuity of γ J γ 7→ (Lemma 3.4-2), which is a consequence of potentialtheory. The key to investigatingthe dynamics of semi-hyperbolic polynomial semigroups is, the continuity of the map γ J (this is highly γ 7→ nontrivial;see[27])andtheJohnnessofthebasinA ofinfinity(cf. [30]). Notethatthecontinuity γ of the map γ J does not hold in general, if we do not assume semi-hyperbolicity. Moreover, γ 7→ one of the original aspects of this paper is the idea of “combining both the theory of rational semigroups and that of random complex dynamics”. It is quite natural to investigate both fields simultaneously. However, no study thus far has done so. Furthermore,inSection5,weprovideawayofconstructingexamplesofpostcriticallybounded polynomialsemigroupswithsomeadditionalproperties(disconnectednessofJuliaset,semi-hyperbolicity, hyperbolicity,etc.) (cf. Lemma5.1,5.2,5.4,5.5,5.6). Byusingthis,wewillseehoweasilysituation (II) above occurs, and we obtain many examples of situation (II) above. AsweeseeinExample1.4andSection5,itisnotdifficulttoconstructmanyexamples,itisnot difficult to verify the hypothesis “postcritically bounded”, and the class of postcritically bounded polynomial semigroups is very wide. Throughout the paper, we will see some phenomena in polynomial semigroups or random dynamics ofpolynomialsthatdo notoccurinthe usualdynamics ofpolynomials. Moreover,those phenomena and their mechanisms are systematically investigated. In Section 2, we present the main results of this paper. We give some tools in Section 3. The proofs of the main results are given in Section 4. In Section 5, we present many examples. There are many applications of the results of postcritically bounded polynomial semigroups in manydirections. Insubsequentpapers[39,40],wewillinvestigateMarkovprocessonCˆ associated with the random dynamics of polynomials and we will consider the probability T (z) of tending to Cˆ starting with the initial value z Cˆ. It will be shown in [39, 40] that i∞f the associated ∞ ∈ ∈ polynomial semigroup G is postcritically bounded and the Julia set is disconnected, then the function T definedonCˆ hasmanyinterestingpropertieswhicharesimilartothose ofthe Cantor function. F∞or example, under certain conditions, T is continuous on Cˆ, varies precisely on J(G) ∞ which is a thin fractal set, and T has a kind of monotonicity. Such a kind of “singular functions ∞ on the complex plane” appear very naturally in random dynamics of polynomials and the study of the dynamics of postcritically polynomial semigroups are the keys to investigating that. (The above results have been announced in [34, 35].) Moreover,as illustratedbefore, it is very importantfor us to recallthat the complex dynamics canbeappliedtodescribesomemathematicalmodels. Forexample,thebehaviorofthepopulation of a certain species can be described as the dynamical systems of a polynomial h such that h preserves the unit interval and the postcritical set in the plane is bounded. When one considers such a model, it is very natural to consider the random dynamics of polynomials with bounded 4 postcritical set in the plane (see Example 1.4). In [37], we investigate the dynamics of postcritically bounded polynomial semigroups G which is possibly generated by a non-compact family. The structure of the Julia set is deeply studied, and for such a G with disconnected Julia set, it is shown that J(G) C, and that if A and B are ⊂ two connected components of J(G), then one of them surrounds the other. Therefore the space of all connected components of J(G) has an intrinsic total order. Moreover, we show that for G eJach n N , there exists a finitely generated postcritically bounded polynomial semigroup 0 ∈ ∪{ℵ } G such that the cardinality of the space of all connected components of J(G) is equal to n. In [38], by using the results in [37], we investigate the fiberwise (random) dynamics of polynomials whichareassociatedwithapostcritically boundedpolynomialsemigroupG.We willpresentsome sufficient conditions for a fiberwise Julia set to be a Jordancurve but not a quasicircle. Moreover, we will investigate the limit functions of the fiberwise dynamics. In the subsequent paper [24], we will give some further results on postcritically bounded polynomial semigroups, based on [37] and this paper. Moreover, in the subsequent paper [36], we will define a new kind of cohomology theory, in order to investigate the action of finitely generated semigroups, and we will apply it to the study of the dynamics of postcritically bounded polynomial semigroups. Acknowledgement: The author thanks R. Stankewitz for many valuable comments. 2 Main results In this section we present the statements of the main results. The proofs are given in Section 4. In order to present the main results, we need some notations and definitions. Definition2.1. WesetRat: = h:Cˆ Cˆ h is a non-constant rational map endowedwiththe { → | } distance η which is defined by η(h ,h ) := sup d(h (z),h (z)), where d denotes the spherical 1 2 z Cˆ 1 2 distance on Cˆ. We set Poly := h : Cˆ Cˆ h ∈is a non-constant polynomial endowed with the { → | } relative topology from Rat. Moreover, we set Poly := g Poly deg(g) 2 endowed with deg 2 ≥ { ∈ | ≥ } the relative topology from Rat. Remark 2.2. Let d 1, pn n N a sequence of polynomials of degree d, and p a polynomial. ≥ { } ∈ Then, p p in Poly if and only if the coefficients converge appropriately and p is of degree d. n → Definition 2.3. For a polynomial semigroup G, we set Kˆ(G):= z C g(z) is bounded in C { ∈ | { } } g[G ∈ and callKˆ(G) the smallest filled-in Julia set ofG. For a polynomialg,we setK(g):=Kˆ( g ). h i Definition 2.4. For a set A Cˆ, we denote by int(A) the set of all interior points of A. ⊂ Definition 2.5 ([27, 30]). 1. Let X be a compact metric space, g :X X a continuous map, and f :X Cˆ X Cˆ a → × → × continuousmap. We saythatf isarationalskewproduct(orfiberedrationalmapontrivial bundle X Cˆ) over g :X X, if π f =g π where π :X Cˆ X denotes the canonical × → ◦ ◦ × → opfrofjeicstiaonn,oann-dcoinf,stfoarntearcahtixon∈alXm,atph,eurnesdterricttihoencfaxn:o=nifca|πl−i1d(e{nxt}i)fi:cπat−io1(n{xπ})1→( xπ−)1(={gC(ˆx)fo}r) − { ′} ∼ each x X. Let d(x) = deg(f ), for each x X. Let f be the rational map defined by: ′ x x,n ∈ ∈ f (y) = π (fn(x,y)), for each n N,x X and y Cˆ, where π : X Cˆ Cˆ is the x,n Cˆ Cˆ ∈ ∈ ∈ × → projection map. Moreover, if f is a polynomial for each x X, then we say that f :X Cˆ X Cˆ is a x,1 ∈ × → × polynomial skew product over g :X X. → 5 N 2. LetΓbe acompactsubsetofRat. We setΓ := γ =(γ ,γ ,...) j,γ Γ endowedwith 1 2 j the product topology. This is a compact metric{space. Let σ : Γ|N∀ ΓN∈be}the shift map, which is defined by σ(γ ,γ ,...) := (γ ,γ ,...). Moreover, we defin→e a map f : ΓN Cˆ 1 2 2 3 ΓN Cˆ by: (γ,y) (σ(γ),γ (y)), where γ =(γ ,γ ,...). This is calledthe skew pr×odu→ct 1 1 2 × 7→ associated with the family Γ of rational maps. Note that f (y)=γ γ (y). γ,n n 1 ◦···◦ Remark 2.6. Let f : X Cˆ X Cˆ be a rational skew product over g : X X. Then, the × → × → function x d(x) is continuous on X. 7→ Definition 2.7 ([27, 30]). Let f : X Cˆ X Cˆ be a rational skew product over g : X X. × → × → Then, we use the following notation. 1. For each x∈X and n∈N, we set fxn :=fn|π−1({x}) :π−1({x})→π−1({gn(x)})⊂X×Cˆ. 2. Foreachx X,wedenote byF (f)the setofpointsy Cˆ whichhavea neighborhoodU in x ∈ ∈ Cˆ suchthat fx,n :U Cˆ n N isnormal. Moreover,wesetFx(f):= x Fx(f)( X Cˆ). { → } ∈ { }× ⊂ × 3. Foreachx X,wesetJ (f):=Cˆ F (f).Moreover,wesetJx(f):= x J (f)( X Cˆ). x x x ∈ \ { }× ⊂ × These sets Jx(f) and J (f) are called the fiberwise Julia sets. x 4. We set J˜(f):= Jx(f), where the closure is taken in the product space X Cˆ. x X × ∈ S 5. For each x X, we set Jˆx(f):=π 1( x ) J˜(f). Moreover,we set Jˆ (f):=π (Jˆx(f)). − x Cˆ ∈ { } ∩ 6. We set F˜(f):=(X Cˆ) J˜(f). × \ Remark 2.8. WehaveJˆx(f) Jx(f)andJˆ (f) J (f).However,strictcontainmentcanoccur. x x Forexample,leth beapolyno⊃mialhavingaSiege⊃ldiskwithcenterz C.Leth beapolynomial 1 1 2 such that z is a repelling fixed point of h . Let Γ= h ,h . Let f :Γ∈ Cˆ Γ Cˆ be the skew 1 2 1 2 product associated with the family Γ. Let x=(h ,h{,h ,..}.) ΓN. The×n, (x→,z )× Jˆx(f) Jx(f) 1 1 1 1 and z Jˆ (f) J (f). ∈ ∈ \ 1 x x ∈ \ Definition 2.9. Let f : X Cˆ X Cˆ be a polynomial skew product over g : X X. Then for each x X, we set×Kx(f→) := ×y Cˆ fx,n(y) n N is bounded in C , and Ax(→f) := y Cˆ f (y)∈ , n . Mor{eov∈er, w|e{set Kx}(f∈) := x K (f) }( X Cˆ) and x,n x { ∈ | → ∞ → ∞} { } × ⊂ × Ax(f):= x A (f) ( X Cˆ). x { }× ⊂ × Definition 2.10. Let f :X Cˆ X Cˆ be a rational skew product over g :X X. We set × → × → C(f):= (x,y) X Cˆ y is a critical point of f . x,1 { ∈ × | } Moreover,we set P(f):= fn(C(f)), where the closure is taken in the product space X Cˆ. n N × This P(f) is called the fibSer-∈postcritical set of f. We say that f is hyperbolic (along fibers) if P(f) F(f). ⊂ Definition 2.11 ([27]). Let f :X Cˆ X Cˆ be a rational skew product over g :X X. Let × → × → N N. We say that a point (x ,y ) X Cˆ belongs to SH (f) if there exists a neighborhood 0 0 N U o∈f x in X and a positive number δ∈such×that for any x U, any n N, any x g n(x), and 0 n − ∈ ∈ ∈ any connected component V of (f ) 1(B(y ,δ)), deg(f : V B(y ,δ)) N. Moreover, xn,n − 0 xn,n → 0 ≤ we set UH(f) := (X Cˆ) N NSHN(f). We say that f is semi-hyperbolic (along fibers) if UH(f) F˜(f). × \ ∪ ∈ ⊂ Remark 2.12. LetΓbeacompactsubsetofRatandletf :ΓN Cˆ ΓN Cˆ betheskewproduct × → × associated with Γ. Let G be the rational semigroup generated by Γ. Then, by Lemma 3.5-1, it is easy to see thatf is semi-hyperbolicif and only if G is semi-hyperbolic. Similarly, it is easy to see that f is hyperbolic if and only if G is hyperbolic. 6 Definition 2.13. Let K 1. A Jordan curve ξ in Cˆ is said to be a K-quasicircle, if ξ is the image of S1( C) under a≥K-quasiconformal homeomorphism ϕ:Cˆ Cˆ. (For the definition of a ⊂ → quasicircle and a quasiconformal homeomorphism, see [15].) Definition 2.14. LetV be asubdomainofCˆ suchthat∂V C.WesaythatV isaJohndomain ⊂ if there exists a constantc>0 anda pointz V (z = when V) satisfying the following: 0 0 ∈ ∞ ∞∈ for all z V there exists an arc ξ V connecting z to z such that for any z ξ, we have 1 1 0 ∈ ⊂ ∈ min z a a ∂V cz z . (Note: in this paper, if we consider a John domain V, we 1 requ{i|re−tha|t|∂V ∈ C.}H≥owe|ver−, in|the original notion of John domain, more general concept of John domains V ⊂was given, without assuming ∂V C ([17]).) ⊂ Remark 2.15. Let V be a simply connected domain in Cˆ such that ∂V C. It is well-known ⊂ thatifV isaJohndomain,then∂V islocallyconnected([17,page26]). Moreover,aJordancurve ξ C is a quasicircle if and only if both components of Cˆ ξ are John domains ([17, Theorem ⊂ \ 9.3]). Definition 2.16. Let X be a complete metric space. A subset A of X is said to be residual if X A is a countable union of nowhere dense subsets of X. Note that by Baire Category Theorem, \ a residual set A is dense in X. Definition 2.17. For any connected sets K and K in C, “K K ” indicates that K = K , 1 2 1 2 1 2 or K is included inabounded componentofC K .Furthermore,≤“K <K ” indicates K K 1 2 1 2 1 2 \ ≤ and K = K . Note that “ ” is a partial order in the space of all non-empty compact connected 1 2 sets in C6. This “ ” is calle≤d the surrounding order. ≤ Letτ beaBorelprobabilitymeasureonPoly .Weconsidertheindependentandidentically deg 2 distributed (abbreviated by i.i.d.) random dynam≥ics on Cˆ such that at every step we choose a polynomial map h : Cˆ Cˆ according to the distribution τ. (Hence, this is a kind of Markov → process on Cˆ. ) Definition2.18. ForaBorelprobabilitymeasureτ onPoly ,wedenotebyΓ thetopological deg 2 τ ≥ support of τ in Poly . (Hence, Γ is a closed set in Poly .) Moreover,we denote by τ˜ the deg 2 τ deg 2 ≥ ≥ N infiniteproductmeasure τ.ThisisaBorelprobabilitymeasureonΓ .Furthermore,wedenote ⊗∞j=1 τ by G the polynomial semigroup generated by Γ . τ τ We present a result on compactly generated, semi-hyperbolic, polynomial semigroups in . G Theorem 2.19. Let Γ be a non-empty compact subset of Poly . Let f :ΓN Cˆ ΓN Cˆ be deg 2 ≥ × → × the skew product associated with the family Γ of polynomials. Let G be the polynomial semigroup generated by Γ. Suppose that G and that G is semi-hyperbolic. Then, exactly one of the ∈ G following three statements 1, 2, and 3 holds. 1. G is hyperbolic. Moreover, there exists a constant K 1 such that for each γ ΓN, J (f) is γ ≥ ∈ a K-quasicircle. 2. There exists a residual Borel subset of ΓN such that, for each Borel probability measure τ U on Poly with Γ = Γ, we have τ˜( ) = 1, and such that, for each γ , J (f) is a deg 2 τ γ ≥ U ∈ U Jordan curve but not a quasicircle, A (f) is a John domain, and the bounded component of γ F (f) is not a John domain. Moreover, there exists a dense subset of ΓN such that, for γ each γ , J (f) is not a Jordan curve. Furthermore, there exist twVo elements α,β ΓN γ such tha∈tVJ (f)<J (f). (Remark: by Lemma 3.6, for each ρ ΓN, J (f) is connected∈.) β α ρ ∈ 3. There exists a dense subset of ΓN such that for each γ , J (f) is not a Jordan curve. γ Moreover, for each α,β ΓNV, J (f) J (f)= . Furtherm∈oVre, J(G) is arcwise connected. α β ∈ ∩ 6 ∅ 7 Corollary 2.20. Let Γ be a non-empty compact subset of Poly . Let f :ΓN Cˆ ΓN Cˆ be deg 2 ≥ × → × the skew product associated with the family Γ of polynomials. Let G be the polynomial semigroup generated by Γ. Suppose that G and that G is semi-hyperbolic. Then, either statement 1 or dis ∈G statement2inTheorem2.19holds. Inparticular, foranyBorelProbability measureτ onPoly deg 2 with Γ =Γ, for almost every γ ΓN with respect to τ˜, J (f) is a Jordan curve. ≥ τ ∈ τ γ We now classify compactly generated, hyperbolic, polynomial semigroups in . G Theorem 2.21. Let Γ be a non-empty compact subset of Poly . Let f :ΓN Cˆ ΓN Cˆ be deg 2 ≥ × → × the skew product associated with the family Γ. Let G be the polynomial semigroup generated by Γ. Suppose that G and that G is hyperbolic. Then, exactly one of the following three statements ∈G 1, 2, and 3 holds. 1. There exists a constant K 1 such that for each γ ΓN, J (f) is a K-quasicircle. γ ≥ ∈ N 2. There exists a residual Borel subset of Γ such that, for each Borel probability measure τ U on Poly with Γ = Γ, we have τ˜( ) = 1, and such that, for each γ , J (f) is a deg 2 τ γ ≥ U ∈ U Jordan curve but not a quasicircle, A (f) is a John domain, and the bounded component of γ F (f) is not a John domain. Moreover, there exists a dense subset of ΓN such that, for γ each γ , J (f) is a quasicircle. Furthermore, there exists a denseVsubset of ΓN such γ ∈ V W that, for each γ , there are infinitely many bounded connected components of F (f). γ ∈W 3. For each γ ΓN, there are infinitely many bounded connected components of F (f). More- γ over, for eac∈h α,β ΓN, J (f) J (f)= . Furthermore, J(G) is arcwise connected. α β ∈ ∩ 6 ∅ Example 2.22. Let g (z):=z2 1 and g (z):= z2. Let Γ:= g2,g2 . Let f :ΓN Cˆ ΓN Cˆ 1 − 2 4 { 1 2} × → × betheskewproductassociatedwithΓ.Moreover,letGbethepolynomialsemigroupgeneratedby Γ. Let D := z C z <0.4 .Then, it is easy to see g2(D) g2(D) D. Hence, D F(G). Let { ∈ || | } 1 ∪ 2 ⊂ ⊂ U be a small disk around 1. Then g2(U) U and g2(U) D. Therefore U F(G). Moreover, − 1 ⊂ 2 ⊂ ⊂ by Remark 1.3, we have that P (G) = g( 0, 1 ) D U F(G). Hence, G ∗ g G Id { − } ⊂ ∪ ⊂ ∈ G and G is hyperbolic. Furthermore, let KS:=∈ z∪{ }C 0.4 z 4 . Then, it is easy to see that { ∈ | ≤ | | ≤ } (g2) 1(K) (g2) 1(K) K and (g2) 1(K) (g2) 1(K) = . Combining this with Lemma 3.1-6 1 − ∪ 2 − ⊂ 1 − ∩ 2 − ∅ and Lemma 3.1-2, we obtainthat J(G) is disconnected. Therefore,G . Leth :=g2 for each ∈Gdis i i i = 1,2. Let 0 < p ,p < 1 with p +p = 1. Let τ := 2 p δ . Since J(g2) is not a Jordan 1 2 1 2 i=1 i hi 1 curve, from Theorem 2.21, it follows that for almost evePry γ ΓN with respect to τ˜, Jγ(f) is a ∈ Jordan curve but not a quasicircle, and A (f) is a John domain but the bounded component of γ F (f)isnotaJohndomain. (See figure1: the JuliasetofG.) Inthis example,foreachconnected γ component J of J(G), there exists a unique γ ΓN such that J =J (f). γ ∈ Example 2.23. Let h (z) := z2 1 and h (z) := az2, where a C with 0 < a < 0.1. Let 1 2 − ∈ | | Γ:= h ,h .Moreover,letG:= h ,h .LetU := z <0.2 .Then,itiseasytoseethath (U) 1 2 1 2 2 U, h{(h (U)}) U,andh2(U) Uh.Hencie,U F(G{)|.I|tfollow}sthatP (G) int(Kˆ(G)) F(G⊂). 2 1 ⊂ 1 ⊂ ⊂ ∗ ⊂ ⊂ Therefore, G and G is hyperbolic. Since J(h ) is not a Jordan curve and J(h ) is a Jordan 1 2 curve, Theore∈mG2.21 implies that there exists a residual subset of ΓN such that, for each Borel U probabilitymeasureτ onPoly withΓ =Γ,wehaveτ˜( )=1,andsuchthat,foreachγ , deg 2 τ ≥ U ∈U J (f) is a Jordan curve but not a quasicircle. Moreover,for each γ , A (f) is a John domain, γ γ ∈U but the bounded component of F (f) is not a John domain. γ Remark 2.24. Let h Poly be a polynomial. Suppose that J(h) is a Jordan curve but deg 2 not a quasicircle. Then∈, it is eas≥y to see that there exists a parabolic fixed point of h in C and the bounded connected component of F(h) is the immediate parabolic basin. Hence, h is not h i semi-hyperbolic. Moreover,by [5], F (h) is not a John domain. ∞ Thus what we see in statement 2 in Theorem 2.19 and statement 2 in Theorem 2.21, as illus- tratedinExample 2.22andExample2.23(seealsoSection5),isaphenomenonwhichcanholdin the random dynamics of a family of polynomials, but cannot hold in the usual iteration dynamics 8 Figure 1: The Julia set of G = g2,g2 , where g (z) := z2 1,g (z) := z2. For a.e.γ, J (f) is a h 1 2i 1 − 2 4 γ Jordancurvebutnotaquasicircle,A (f)isaJohndomain,andthe boundedcomponentofF (f) γ γ isnotaJohndomain. ForeachconnectedcomponentJ ofJ(G), thereexistsauniqueγ ΓN such ∈ that J =J (f). γ of a single polynomial. Namely, it can hold that for almost every γ ΓN, J (f) is a Jordan curve γ ∈ and fails to be a quasicircle all while the basin of infinity A (f) is still a John domain. Whereas, γ if J(h), for some polynomial h, is a Jordan curve which fails to be a quasicircle, then the basin of infinity F (h) is necessarily not a John domain. ∞ InSection5,wewillseehoweasilysituation2inTheorem2.19andsituation2inTheorem2.21 occur. Pilgrim and Tan Lei ([18]) showed that there exists a hyperbolic rational map h with discon- nected Julia set such that “almost every” connected component of J(h) is a Jordancurve but not a quasicircle. We give a sufficient condition so that statement 1 in Theorem 2.21 holds. Proposition 2.25. Let Γ be a non-empty compact subset of Poly . Let f :ΓN Cˆ ΓN Cˆ deg 2 ≥ × → × be the skew product associated with the family Γ. Let G be the polynomial semigroup generated by Γ. Suppose that P (G) is included in a connected component of int(Kˆ(G)). Then, there exists a ∗ constant K 1 such that for each γ ΓN, J (f) is a K-quasicircle. γ ≥ ∈ Example 2.26. Let d1,...,dm N with dj 2 for each j, and let hj(z) = ajzdj +cj,aj = 0, ∈ ≥ 6 for each j = 1,...,m. Let Γ = g ,...,g . If c is small enough for each j, then Γ satisfies the 1 m j h i | | assumption of Proposition 2.25. Thus statement 1 in Theorem 2.21 holds. We have also many examples of Γ such that statement 3 in Theorem 2.19 or statement 3 in Theorem 2.21 holds. Example 2.27. Let h Poly . Suppose that h and h is hyperbolic. Suppose also thath hasatleasttwo1a∈ttractindegg≥p2eriodicpointsinhC.1Lie∈tΓGbe asm1allcompactneighborhoodof 1 h in Poly . Then Γ and Γ is hyperbolic (see Lemma 5.4). Moreover,by the argument 1 deg 2 in the proofo≥fLemmah5.6i,∈wGesee thhaitfor eachγ ΓN,F (f) hasatleasttwo boundedconnected γ ∈ components, where f :ΓN Cˆ ΓN Cˆ is the skewproduct associatedwith Γ. Thus statement3 × → × inTheorem2.21holds. WeremarkthatbyusingLemma5.5,5.6andtheirproofs,weeasilyobtain manyexamplesofΓsuchthatstatement3inTheorem2.19orstatement3inTheorem2.21holds. 3 Tools To show the main results, we need some tools in this section. 9 3.1 Fundamental properties of rational semigroups Notation: For a rational semigroup G, we set E(G):= z Cˆ ♯( g 1( z ))< . This is { ∈ | g G − { } ∞} called the exceptional set of G. S ∈ Notation: Let r > 0. For a subset A of Cˆ, we set B(A,r) := z Cˆ d (z,A)< r , where d is s s the spherical distance. For a subset A of C, we set D(A,r) :={z∈ C| d (z,A)<r}, where d is e e { ∈ | } the Euclidean distance. We use the following Lemma 3.1 and Theorem 3.2 in the proofs of the main results. Lemma 3.1 ([12, 11, 29, 27]). Let G be a rational semigroup. 1. For each h G, we have h(F(G)) F(G) and h 1(J(G)) J(G). Note that we do not − ∈ ⊂ ⊂ have that the equality holds in general. 2. If G = hh1,...,hmi, then J(G) = h−11(J(G)) ∪···∪h−m1(J(G)). More generally, if G is generatedbyacompact subsetΓofRat, thenJ(G)= h 1(J(G)).(Wecallthis property h Γ − of the Julia set of a compactly generated rational semSigr∈oup “backward self-similarity.” ) 3. If ♯(J(G)) 3 , then J(G) is a perfect set. ≥ 4. If ♯(J(G)) 3 , then ♯(E(G)) 2. ≥ ≤ 5. If a point z is not in E(G), then J(G) g 1( z ). In particular if a point z belongs ⊂ g G − { } ∈ to J(G) E(G), then g 1( z )=JS(G). \ g G − { } ∈ S 6. If ♯(J(G)) 3 , then J(G) is the smallest closed backward invariant set containing at ≥ least three points. Here we say that a set A is backward invariant under G if for each g G, g 1(A) A. − ∈ ⊂ Theorem 3.2 ([12, 11, 29]). Let G be a rational semigroup. If ♯(J(G)) 3, then ≥ J(G) = z C g G, g(z)=z, g (z) >1 , where the closure is taken in Cˆ. In particular, { ∈ |∃ ∈ | ′ | } J(G)= J(g). g G ∈ S Remark 3.3. If a rationalsemigroupG containsan element g with deg(g) 2,then ♯(J(g)) 3, ≥ ≥ which implies that ♯(J(G)) 3. ≥ 3.2 Fundamental properties of fibered rational maps Lemma 3.4. Let f :X Cˆ X Cˆ be a rational skew product over g :X X. Then, we have × → × → the following. 1. ([27, Lemma 2.4]) For each x X, (f ) 1(J (f)) = J (f). Furthermore, we have x,1 − g(x) x ∈ Jˆ (f) J (f). Note that equality Jˆ(f)=J (f) does not hold in general. x x x x ⊃ If g :X X is a surjective and open map, then f 1(J˜(f))=J˜(f)=f(J˜(f)), and for each − x X, (→f ) 1(Jˆ (f))=Jˆ (f). x,1 − g(x) x ∈ 2. ([14, 27]) If d(x) 2 for each x X, then for each x X, J (f) is a non-empty perfect set x ≥ ∈ ∈ with ♯(J (f)) 3. Furthermore, the map x J (f) is lower semicontinuous; i.e., for any x x ≥ 7→ point (x,y) X Cˆ with y Jx(f) and any sequence xn n N in X with xn x, there exists a sequ∈ence×yn n N in∈Cˆ with yn Jxn(f) for each{n } N∈ such that yn →y. However, { } ∈ ∈ ∈ → x J (f) is not continuous with respect to the Hausdorff topology in general. x 7→ 3. If d(x) 2 for each x X, then inf diam J (f)>0, where diam denotes the diameter x X S x S ≥ ∈ ∈ with respect to the spherical distance. 10