ADDING MACHINES AND WILD ATTRACTORS

We investigate the dynamics of unimodal maps f of the interval restric ted to the omega limit set X of the critical point for cases where X i s a Canter set. In particular, many cases where X is a measure attract or of f are included. We give two classes of examples of such maps, bo th generalizing unimodal Fibonacci maps [LM, BKNS]. In all cases f(\X) is a continuous factor of a generalized odometer (an adding machine-l ike dynamical system), and at the same time f(\X) factors onto an irra tional circle rotation. In some of the examples we obtain irrational r otations on more complicated groups as factors.