This monograph examines rotation sets under the multiplication by d (mod 1) map and their
relation to degree d polynomial maps of the complex plane. These sets are higher-degree analogs
of the corresponding sets under the angle-doubling map of the circle which played a key role
in Douady and Hubbard's work on the quadratic family and the Mandelbrot set. Presenting the
first systematic study of rotation sets treating both rational and irrational cases in a
unified fashion the text includes several new results on their structure their gap dynamics
maximal and minimal sets rigidity and continuous dependence on parameters. This abstract
material is supplemented by concrete examples which explain how rotation sets arise in the
dynamical plane of complex polynomial maps and how suitable parameter spaces of such
polynomials provide a complete catalog of all such sets of a given degree. As a main
illustration the link between rotation sets of degree 3 and one-dimensional families of cubic
polynomials with a persistent indifferent fixed point is outlined. The monograph will benefit
graduate students as well as researchers in the area of holomorphic dynamics and related
fields.
