A Characterization of Attractors for Baire Functions on the Interval

Abstract

We show that for any closed set $E\subset \lbrack 0,1]$, there exists $f:[0,1]\rightarrow \lbrack 0,1]$ in the first class of Baire that generates $E$ as one of its $\omega $-limit sets. Since $\omega $-limit sets are necessarily closed, this characterizes the class of $\omega $-limit sets for Baire or Borel functions on the interval.