Sets of recurrent points of continuous maps of the interval

Abstract

For a continuous map of the interval the following conditions are equivalent: (1) the period of every periodic point is a power of 2, (2) R ¯ ( + ) ∩ R ¯ ( − ) − R = ϕ {\overline R ^{( + )}} \cap {\overline R ^{( - )}} - R = \phi , and (3) R ¯ − R \overline R - R is countable, where R R denotes the set of recurrent points. R ¯ \overline R is the closure of R R , and R ¯ ( + ) {\overline R ^{( + )}} (or R ¯ ( − ) {\bar R^{( - )}} ) is the right-side closure (left-side closure, respectively) of R R .