Reversibility in Groups of Piecewise Linear Homeomorphisms of the Circle

Let PL ( S ) be the group of piecewise linear homeomorphisms of the circle S which are locally affine at all but a finite number of points of S (called break points) and PL + ( S ) be the subgroup of PL ( S ) whose elements are order preserving. An element in PL + ( S ) is called reversible in PL + ( S ) if it is conjugate to its inverse in PL + ( S ) . In this paper, we answer partially one of O’Farrell and Short’s problems in their recent book (O’Farrell and Short in Reversibility in dynamics and group theory. London Mathematical Society Lecture Note Series: 416. Cambridge University Press, Cambridge, 2014). We characterize the reversible elements in PL + ( S ) and also perform a similar characterization in the full group PL ( S ) . In particular, we give a complete characterization of reversibility in PL + ( S ) and PL ( S ) of elements f of PL + ( S ) having the ( D )-property (i.e. for which the product of the f -jumps along the full orbit of any break point is one).