NECESSARY AND SUFFICIENT CONDITIONS FOR ZERO ENTROPY OF SELF-MAPPINGS OF THE CIRCLE

Let S^1 be a unit circle. For a, b∈S^1, a≠b, (a, b] and [a, b) denote thehalfopen arcs from a to b counterclockwise. For f∈C^O(S^1, S^1), the topological entropy of f is denoted by ent(f). R(f) is the set of all recurrent points of f, and is its closure.