Chaotic behavior in piecewise continuous difference equations

Abstract

A class of piecewise continuous mappings with positive slope, mapping the unit interval into itself is studied. Families of 1-1 mappings depending on some parameter have periodic orbits for most parameter values, but have an infinite invariant set which is a Cantor set for a Cantor set of parameter values. Mappings which are not 1-1 exhibit chaotic behavior in that the asymptotic behavior as measured by the rotation number covers an interval of values. The asymptotic behavior depends sensitively on initial data in that the rotation number is either a nowhere continuous function of initial data, or else it is a constant on all but a Cantor set of the unit interval.