Extending circle mappings to the annulus (Q1863017)

It is shown that given a monotone \(C^\infty\) map \(A\) of the circle, there is a continuous map \(h\) from the half-open annulus \(S^1\times [0,\infty)\) to itself such that (i) \(h(x,0) = (A(x),0)\) for all \(x\in S^1\); (ii) on the open annulus \(S^1\times (0,\infty)\) \(h\) is an area preserving \(C^\infty\) diffeomorphism; and (iii) \(h\) satisfies a twist condition: if we write \(h(x,y) = (h_1(x,y),h_2(x,y))\), then \(\partial h_1/\partial y \neq 0\) on the open annulus. If in addition \(A\) is either a homeomorphism or a diffeomorphism, then so is \(h\).