Let G be the group of germs of
Ck local homeomorphisms of the real line which fix the origin and have nonzero
derivative there. In this paper the possibility of factoring an element of G which is
conjugate to its inverse into the product of two involutions is investigated. It is shown
that it is always possible to do this in the analytic case and not always possible in the
continuous case. In the intermediate cases several necessary and sufficient
conditions are developed for determining whether or not such a factorization is
possible. Included is a construction which allows one to determine an explicit
factorization. Indication is given of the application of this material to the
same problem in higher dimensions. This work is related to some material in
Dynamics.
