Periodic Orbits of Maps on the Disk With Zero Entropy

Abstract

Let M be a set and f a map from M into itself. In this quite general setting, one can already define the orbit of a point x in M as the set {x, ƒ(x), ƒ²(x),...} where ƒⁿ stands for ƒ ΰ ƒ^n-1, ƒ⁰ being the identity map Id_M. Assume that for some x⁰ in M, there is a n > 0 such that ƒⁿ(x₀) = x₀. Then there is a smaller such n, say p = min{n > 0 such that ƒⁿ(x₀) = x₀}, and we will say that x₀ is a periodic point (of period p) and that the set {x, ƒ(x), ƒ²(x),..., ƒ^p-1(x₀)} is a periodic orbit.