1 Entropy of Convolutions on the Circle , I after

Let p ≥ 2 be any integer (p need not be prime), and T = R/Z the 1-torus. Denote by σp the p-to-one map x 7→ px (mod 1). The pair (T, σp) is a dynamical system that has additional structure: T is a commutative group (with the group operation being addition mod 1), and σp is an endomorphism of it. Even in such a simple system, the interaction between the dynamics and the algebraic structure of T can be quite subtle; the present work continues the study of this interaction, inspired by the fundamental work of Furstenberg [1].