One-parameter transformation groups in the plane

Abstract

PROOF. Let xEE. Since x is not fixed under R, there is a closed interval [-a, a ] = T about 0 in R, and an arc CCE, xE C but not an end point of C, such that T2(C) is a compact neighborhood of x and the mapping (t, c)->t(c) is one to one from T2X C-->T2(C). That is, C is a local cross section to the local orbits of T2 [1]. We shall show that C is a local cross section for the orbits of R. Suppose, on the contrary, that for some zECC, there is an r>a such that r(z)EzT(C). Let b be the greatest lower bound of such numbers. Then b(z)E-a(C), for if not, say b(z)=t(c), tGET, cCC, and t >-a, then there is a t', -a -a. Hence (b+t')(z)=(t+t')(c)CET(C). But t'<O so that b+t'<b. By the choice of b, this implies b+t'< a. Since this implies b+t <a for