Sharkovskĭi Theorem for Multidimensional Perturbations of One-dimensional Maps Ii

We present a multidimensional generalization of the Sharkovskĭı Theorem concerning the appearance of periodic points for the self-maps on the real line. Introduction Let f : X → X be a map. We say that x ∈ X is a periodic point of period n if f(x) = x and f(x) 6= x for k = 1, . . . , n− 1. We begin with recalling the Sharkovskĭı Theorem. Theorem 1. Let the ordering of positive integers be as follows: 3 C 5 C 7 C . . . C 2 · 3 C 2 · 5 C 2 · 7 C . . . C 22 · 3 C 22 · 5 C . . . C 23 · 3 C 23 · 5 C . . . C 23 C 22 C 2 C 1. Let f : I → R be a continuous map of an interval into the real line. If n C k and f has a periodic point of period n then f also has a periodic point of period k. The ordering described in Theorem 1 is called the Sharkovskĭı ordering. We reserve the symbol “C” for this order. 1991 Mathematics Subject Classification. 37E05, 37L99.