Abstract

We consider a family of piecewise contractions admitting a rotation number and defined for every x is an element of[0,1) by f(x)=lambda x+delta+d theta a(x)(mod1), where lambda is an element of(0,1), d is an element of(0,1-lambda), delta is an element of[0,1], a is an element of[0,1] and theta a(x)=1 if x >= a and theta a(x)=0 otherwise. In the special case where a = 1, the family reduces to the well studied 'contracted rotations' x bar right arrow lambda x+delta(mod1), which are 2-interval piecewise lambda-affine contractions when delta is an element of(1-lambda,1). Considering a is an element of(0,1) allows maps with an additional discontinuity, that is, 3-interval piecewise lambda-affine contractions. Supposing lambda and d fixed, for any rho is an element of(0,1) and alpha is an element of[0,1], we provide the values of the parameters delta and a for which the corresponding map has rotation number rho, and a symbolic dynamics containing that of the rotation R rho:[0,1)->[0,1) of angle rho with respect to the partition given by the positions of 1-rho and alpha in [0,1). This enables in particular to determine the maps that have a given number of periodic orbits of an arbitrary period, or a Cantor set attractor supporting a dynamics of a given complexity.