PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 10, October 1997, Pages 3051{3058 S 0002-9939(97)04159-2 NIELSEN-THURSTON REDUCIBILITY AND RENORMALIZATION OLIVIER COURCELLE, JEAN-MARC GAMBAUDO, AND CHARLES TRESSER (Communicated by Linda Keen) Abstract. Consider an orientation preserving homeomorphism f of the 2- disk with an infinite set of nested periodic orbits n n 1, such that, for all {O } ≥ m>1, the restriction of f to the complement of the first m orbits, from 1 O to m,ism 1 times reducible in the sense of Nielsen and Thurston. We defineO combinatorial− renormalization operators for such maps, and study the fixed points of these operators. We also recall the corresponding theory for endomorphisms of the interval, and give elements of comparison of the theories in one and two dimensions. 1. Introduction Thurston's completion [Th] of the work of Nielsen [Ni] on the classification, up to isotopy, of surface homeomorphisms was accomplished at about the same time renormalization group ideas were introduced in the theory of one-dimensional dy- namical systems [Fe], [CT], [TC]. Some key concepts from these theories, which were formulated quite independently, are quite similar. In this paper we examine some relationships between the two theories. Our aim is to discuss some combi- natorial aspects of renormalization group theory for orientation preserving homeo- morphisms of the 2-disk, and compare them with what is known in this context for endomorphisms of the interval. In the following discussion, for a sequence (ai)i 1 of integers greater than or equal to 2, we set ≥ q = a a a . n 1 · 2 ····· n m For f a continuous map of the unit m-dimensional disk D into itself and (ai)i 1 a ≥ sequence as above, we say that f is (ai)i 1-infinitely renormalizable if there exists a sequence of m-disks ≥ m D 1(f) 2(f) n(f)... ⊃D ⊃D ⊃···⊃D such that, for each n, f j( (f)) (f)= , for 0 <j q 1, Dn ∩Dn ∅ ≤ n− and f qn ( (f)) (f). Dn ⊂Dn In such a case, by Brouwer's fixed point Theorem, n contains one point xn of a periodic orbit with period q . Collections of periodicD orbits arising this way have On n Received by the editors October 24, 1995. 1991 Mathematics Subject Classification. Primary 58F99. c 1997 American Mathematical Society 3051 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3052 OLIVIER COURCELLE, JEAN-MARC GAMBAUDO, AND CHARLES TRESSER well understood special combinatorial structures in the special cases when f is an endomorphism and m =1,andwhenfis an orientation preserving homeomorphism qn and m = 2. The map which associates f n(f) to f is called a renormalization operator and plays an important role in many|D problems of smooth and holomorphic dynamics. We shall define renormalization operators in the combinatorial context for the cases of dimensions 1 and 2, and compare their dynamical properties. In particular, we shall see that fixed points of combinatorial renormalization operators, as defined in 3, are much more abundant for endomorphisms in one dimension than for homeomorphisms§ in two dimensions, despite the fact that permutation groups are finite while braid groups are not. 2. Combinatorics There are classical ways to associate combinatorial objects to periodic orbits of endomorphisms of the interval I (or f End(I)) and orientation preserving ∈ + homeomorphisms of the 2-disk D2 (or f Homeo (D2)). If x <x < <x are the p points∈ of a periodic orbit of f End(I), 1 2 ··· p O ∈ let h be the bijection sending xi to i. There is a single permutation π,infacta 1 cycle, in the permutation group Sp on p elements such that f = h− π h.We say that π represents the action of f on ,andthatfrealizes|O the cycle◦ π◦. More general permutations can be used similarlyO to represent the dynamics of f on finite collections of periodic orbits. Assume now that the p points x1,x2,...,xp of a periodic orbit of f + 2 2 O ∈ Homeo (D ) are in the interior of D ,andthaty1,y2,...,yp are the p points of + 2 a periodic orbit 0 of g Homeo (D ). We say ( ,f)and( 0,g) have the same O ∈ + 2 O O 1 braid type if there exists h Homeo (D ) sending onto 0, such that h− g h is isotopic to f relative to ∈. It is straightforward toO checkO that \to have the◦ same◦ braid type" is an equivalenceO relation. The corresponding equivalence classes are called braid types:if( ,f) is a representative of the braid type β,wealsosaythat O βrepresents the action of f on D2 ,andthatfrealizes the braid type β. More general braid types can be used similarly\O to represent the dynamics of f on finite collections of periodic orbits. The braid types with n strands form the braid type set BTn which is obtained from the Artin braid group Bn by quotienting by the conjugacies and the center: in the case of a single periodic orbit, any representative b in Bn of a braid type β in BTn is mapped to a cycle by the canonical projection τ from Bn to Sn:wesayβis a cycle if τ(b) is a cycle (a property which does not depend on the choice of b). m m If b and b0 are two representatives in B of β BT ,thenb and b0 represent n ∈ n the same element of BTn for each m Z. As a consequence, any β BTn ∈ ∈ generates a group Gβ by iteration. Remark 1. The set of powers of a permutation form a cyclic subgroup of Sn.On the contrary, assuming β is a cycle, Gβ is a cyclic group if and only if β is an elementary braid type in the sense of Nielsen-Thurston, i.e., β is the braid type of a periodic orbit of a rigid rotation of the disk. In all other cases when β is a cycle, Gβ is isomorphic to Z. A sequence ( i)i 1 of periodic orbits of f End(I), with periods qi,isan(ai)i 1- cascade of periodicO ≥ orbits if for each j 1, there∈ are pairwise disjoint intervals I≥ ≥ j;k with 0 k qj 1sothateachIj;k contains one point of j and aj+1 points of ≤which≤ are− all mapped to the same I . To each cascadeO of periodic orbits Oj+1 j;l License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NIELSEN-THURSTON REDUCIBILITY AND RENORMALIZATION 3053 of an endomorphism on the interval corresponds a well-defined sequence (πi)i 1 of ≥ permutations, where πi Sqi represents the action of f on i.Wedenoteby the set of such sequences∈ of permutations. O P + 2 A sequence ( i)i 1 of periodic orbits of f Homeo (D ), with periods qi,is O ≥ ∈ an (ai)i 1-cascade of periodic orbits if for each j 1, there are pairwise disjoint disks D≥ with 0 k q 1sothateachD contains≥ one point of and a j;k ≤ ≤ j − j;k Oj j+1 points of j+1,and∂Dj;k is mapped to a curve homotopic to ∂Dj;l relative to the points of O , ,..., .Letβ bethebraidtypeof( ,f): then we denote by O1 O2 Oj+1 l Ol β1 β2 βn thebraidtypeof( 1 2 n,f). To∪ each∪···∪ cascade of periodic orbitsO of an∪O orientation∪···∪O preserving homeomorphism of the 2-disk corresponds a well-defined sequence (βi)i 1 of braid types, where ≥ βi BTqi represents the action of f on i.Wedenoteby the set of such sequences∈ of braid types (for a different approachO to cascades,B see [GGH]). By Remark 1, elements of generate isomorphic copies of Z by iteration. Notice also 2 2 B that if f : D D has an (ai)i 1-cascade of periodic orbits, then, for all m>0, → ≥ the restriction of f to the complement of the first m orbits, from 1 to m,ism 1 times reducible in the sense of Nielsen and Thurston [Ni], [Th]. O O − + 2 Remark 2. Let ( i)i 1 be a cascade of a periodic orbit of f Homeo (D ), and O ≥ ∈ let (βi)i 1 stand for the corresponding sequence of braid types. The braid type ≥ a1 β2 is determined by β2 D2;j . Thus the braid type β1 β2 is determined by the | a1 ∪ choice of a representative b1 of β1 in Ba1 , β D2;j , and an integer k2 Z which 2 | ∈ represents an element of the center of Ba2 . Similarly the braid type β1 β2 β3 a1 q2 ∪ ∪ is determined by b1, β D2;j , β D3;l , and a pair of integers (k2,k3) Z.Ifwe 2 | 3 | ∈ introduce an integer k1 which represents an element of the center of Ba1 by which we conjugate b1 to another representative of β1 in Ba1 ,(k2,k3) has to be replaced by (k2 + k1a2,k3 +k1a2a3). Otherwise speaking, the meaningful index ki is not an integer, but a residue class modulo qi . This allows us to sometimes use the a1 qi 1 sequence ((βi − ,ki))i 1 with k1 = 0 instead of the sequence (βi)i 1 to represent a cascade of periodic orbits.≥ ≥ Clearly, we have Proposition 1. An endomorphism of the interval or an orientation preserving homeomorphism of the 2-disk which is (ai)i 1-infinitely renormalizable has an ≥ (ai)i 1-cascade of periodic orbits. ≥ To facilitate the discussion, it is useful to extend the definition of cascades to include cases when the πi's or the βi's are not necessarily cycles: we just say cascade in the general case, instead of \cascade of periodic orbits".