ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE SHINJI EGASHIRA Expansion growth of foliations Annales de la faculté des sciences de Toulouse 6e série, tome 2, no 1 (1993), p. 15-52 <http://www.numdam.org/item?id=AFST_1993_6_2_1_15_0> © Université Paul Sabatier, 1993, tous droits réservés. L’accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitu- tive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Annales de la Faculte des Sciences de Toulouse Vol. II, n° 1, 1993 Expansion growth of foliations(*) SHINJI EGASHIRA(1) RESUME. - Nous definissons la croissance de 1’elargissement transverse des feuilletages qui peuvent ctre consider~s comme une croissance du car- dinal maximal des ensembles separes d’un pseudo-groupe d’holonomie. Nous prouvons que la croissance de 1’elargissement transverse est un in- variant topologique des feuilletages, et nous calculons la croissance de 1’elargissement transverse de quelques feuilletages typiques de co dimen- sion 1.. ABSTRACT. - We define the expansion growth of foliations which is, roughly speaking, growth of the maximum cardinality of separating sets with respect to a holonomy pseudogroup. We prove that the expansion growth is topological invariant for foliations and we compute the expansion growth of several typical foliations of codimension 1.. 0. Introduction The entropy of foliations is defined by Ghys, Langevin and Walczak Their definition was done by generalizing Bowen’s definition of the entropy of dynamical systems [B]. Let F be a codimension q foliation of class C~ on a compact manifold M. When we fix a finite foliation cover U of ( M, ~’), we obtain the holonomy pseudogroup ?~ of local homeomorphisms of Rq induced by 1.~. Then we can define an integer (n E > 0) to be the maximum cardinality of (n, c)-separating sets with respect to the holonomy pseudogroup ?~ .. sn ( ~ ) is monotone increasing on n and monotone decreasing on e. The entropy of the foliation ~’ is defined by the following formula: ~ * ~ Recu le 12 Novembre 1992 (1)) Department of Mathematical Sciences, University of Tokyo, 7-3-1, Hongo, Bunkyoku, Tokyo, 113 (Japan) - 15 - When we fix a sufficiently small positive real number we notice that the monotone increasing map with respect to n represents the expansion of the foliation. For example, if ~ is a Reeb component then the growth of with respect to n is the linear growth. If ~’ is a linear foliation on then the growth of is the constant growth. In this paper, we consider the growth type of sn (E) defined in the growth set which is an extension of the usual growth set ( cf. [HH2]) and we prove that the growth of depends only on (M, ~’). Therefore it becomes a topological invariant for foliations. Moreover we define an integer to be the minimum cardinality of (n, c)-spanning sets with respect to the holonomy pseudogroup ~C and we show that the growth of is equal to the growth of We call it the expansion growth of (M, ~’). The expansion growth of foliations produces a number of numerical topological invariants for foliations. We also show that many of them are non-trivial invariants. In section 1, we define the growth set which is an extansion of the usual growth set. The expansion growth of foliations is defined as an element of this growth set. In section 2, we define the expansion growth of a foliation and we describe several important properties of the expansion growth. In section 3, we compute the expansion growth of several typical codimension 1 foliations. We also construct foliations which have various expansion growths. 1. Growth In this section, we define the growth of an increasing sequence of incresing functions. Let Z be the set of nonnegative increasing functions on N: Let ~’ be the set of increasing sequences of I. We regard Z as a subset of Z by the map We define the growth type of an element of l. We define a preorder « in as follows. For ’ , (hk) k~IN G ’, In the definition, we note that B F_ IN is independent of j E IN and A > 0 may depends on j E IN . The preorder -~ induces an equivalence relation ^_. in I. We define e to be the set of equivalence classes in I: ~ is the set of all growth types of increasing sequences of increasing functions and has the partial order ~ induced by the preorder . The equivalence written and is called the class of j~IN G T is by j~IN ~ growth type of The equivalence class of g ~ I ~ is simply written by [g]. Let ~* be the set of such growth types: e is equal to the partial ordered set of all growths of monotone increasing functions in the usual sense (cf. [HH2]) and £ can be considered as an extension of it.. For example, the following relation is easily seen Here [0] (resp. [1]) is the growth of the constant function whose value is 0 (resp. 1). [en] E ~ is said to have exactly exponential growth. For k E IN U ~0~_, E ~ is said to have exactly polynomial growth of degree ~. E ~ is said to be quasi-exponential if Next we define the finite sum and the finite product of elements of ~. For E ~~ For example, the following relations are easily seen. 2. Expansion growth of foliations In this section, we define the expansion growth of a foliation on a compact manifold and describe several properties of it. Let ~’ be a C~ codimension q foliation on a compact (p + q)-dimensional manifold M. We assume that aM = 0, however the same discussion is applicable in the case where 0 provided ~’ is tangent or transverse to aM. A = U, is called a triple of foliation covers of (M, ~’~ if it satisfies the following conditions. (1) Ub, U and Us are open coverings of M. (2) Ub = (Ub~ ~~’) A ~ u = and _ , ~pi )}A 1. (3) = - (4) 03C6i 03C6bi |Ui and cpZ . (5) ~~(Ub) = = and = = x Bq1(oi) where o = (o, ... , 0) E IRp, oi (7i, ... , 7i) E IRq, and J3~(~) = {.r e IRP; ~! A;} G !R~. (6) If ut 0, then there exists a local homeomorphism of IRq such that ~b = o ~b on Ub where ~b = pr and pr : IRP x IRq --~ IRq is the projection to the second factor. We put Bi = C IRq. We call the set (~p,;~ 1 (B2 (o~ x ~z~~ (z E Bi~ a plaque of U (or a plaque of Let be the set of all plaques of Lt. Fix a metric d on M. We define the diameter of U to be the maximum = of the diameters of Ui (i 1, ..., , A~: We define and width(U,Ub) as follows: Here for K, K’ ç M, Let Leb(U) denote the Lebesgue number of U. a a Let x be point of M and let n be natural number. Let Pi ..., Pn be to an x x a plaques of U. (Pi, ..., , Pn ) is said be n-chain of Pu at if is point of and n Let be Pi Pl Ø. Ui1, ... foliation neighborhoods ofU. , ... , is said to be an n-chain of U at x if there exists an n-chain ... of Pu at x (uniquely determined) such that is a plaque of . When an n-chain of U at x is the n-chain Uil ... given, of at x with respect to ... , is often written by ... For xl, ..., E M, Let be the set of all n-chains of U at xl and put For K C M, let C be the set of Pn such that there exists ~ E K an at x. and n-chain , ... , Pn ) of Pu For plaques P and P’ of Ui, put Here Pb (resp. is the plaque of Ub containing P (resp. P’). For x, y E M, we define an n-chain of Here is the plaque of where ... , is ~u = . If then at x uniquely determined by , ... E y) ~, y~ = d(x, y). We remark that usually dn is not continuous. However, we can easily show the following lemma. LEMMA 2.1.2014 dn is lower semi-continuous. Let n be a natural number and let e be a positive number. For x, y E M, x and y are said to be (n, c, A)-separated if y~ > ~. Otherwise x and y are said to be (n, c, A)-close. Let K be a subset of M. S ç M is said to be an (n, e, .~4., K)-separating set if 5’ is a subset of K and for any x, y E S, x and y are (n, c, A)-separated. R C M is said to be an (n, ~, r4., K)-spanning set if for any x E K, there exists y E R such that x and yare (n, ~, A)-close. Since M is compact, S is a finite set ( c f. lemma 2.14 and theorem 2.1 S ~ . Put PROPOSITION 2.2 rn (~, K) and K) are monotone increasing on n E IN and monotone decreasing on E > 0.. If K C K’ then rn (e, K) rn (e, K’) and K) sn (E, K~~. Proof. - We only prove that Let S C K be an (n, ~, .~4., K)-separating set with the maximum cardinal- ity.