Behavior of distant maximal geodesies in 2-dimensional manifolds

/(0):=0.

For all i e \n(a)]

11

#-\ _ M +/(' ~ 1) tiBi is not a lemon

A

~ l - / ( i - l ) otherwise,

and for all / n(a) + 1

Then

/(0:=/(i-D.

rot(a) = lim sup \f(i) |.

Proo/. Eliminate lemons by using compactly supported regular homotopies. •

Remark. The proposition clearly implies the fact that for a semi-regular curve a

one has rot(a) ind(a) ~n(a).

a regular curve

an almost regular curve

1.10. Definition, (i) A semi-regular curve a will be called almost regular if

ind(a) = &(ind(a)).

(ii) An almost regular curve will be called regular if in addition ~n{a) = ind(a).