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Takashi SfflOYA

1.2. Definition. Suppose / = R and let n+(a) (resp. n_(a)) be the number

(possibly infinite) of double points having positive (resp. negative) sign of a proper

transversal immersion a: R -» M and let

n(a) = lim sup | n+(sj) - n_(sj) |,

S-+-00

t-*+oo

where n+(s,t) (resp. n_(s,t)) denotes the number of positive (resp. negative)

double points of the closed arc a|[s,r]. Whenever a is such that these three

quantities are not all equal to infinity, the rotation number rot(a) e N U {»} is

defined to be the presently introduced quantity n(a), (where N denotes the set of

nonnegative integers).

Remark. Notice that n(a) (and therefore rot(a) whenever it is defined) does

not depend on the chosen orientation of M, so that the notion of rotation number

makes sense even when M is not assumed to be oriented. Notice also that n(a)

does not depend on the parameterization of a and is an invariant of the compactly

supported regular homotopy class of a. Recall that a proper transversal differentiable

immersion a: R -* M is said to be compactly supported regular homotopic to the

proper transversal immersion j3 when a(t) - p(t) for all t outside some open

interval (a,b) C R and when there exists a regular homotopy between a\[a,b] and

P\[a,b] fixing d{a) and d(b).

1.3. The order relation between double points. Let a be a proper

transversal immersion of some (not necessarily compact) closed interval of R into M

and let p\ and pi be two double points of a such that pt = a(ai) - cc(bi) with

a\ bi for i = 1,2. Set p\ P2 whenever [a\,b\] C [a2,62]- This

convention defines a partial order relation on the set of double points of a such that

for each double point p there exists at least one minimal double point q such that q

p. Notice that, when both the relations p\ P2 andp2 ^p\ do not hold, one of