Dynamics of typical Baire-1 functions on the interval
Let \(B\) denote the set of Baire class 1 functions \([0,1]\to[0,1]\) (that is, pointwise limits of continuous functions), and write as usual \(\omega(x,f)\) for the set of cluster points of the forward orbit of \(x\) under iteration of \(f\in B\). The two main theorems are these. First, there is a residual set \(S\subset B\) with the property that for any function \(f\in S\) the set \(\omega(x,f)\) is contained in the set of continuity points of \(f\) and on \(\omega(x,f)\) the map is an \(\infty\)-adic adding machine, for every \(x\in[0,1]\). Second, there is a residual set \(S'\subset B\) with the property that for any \(f\in S'\) and \(\epsilon>0\) there is some \(M\) for which \(f^m([0,1])\) is contained in the \(\epsilon\)-neighbourhood of \(\bigcup_{x\in[0,1]}\omega(x,f)\) for \(m>M\). Somewhat surprisingly, these results are similar to similar statements known to hold in the more familiar setting of iteration of continuous maps of the interval (see, for example, the paper of \textit{H. Lehning} [Proc. Am. Math. Soc. 123, No. 6, 1703--1707 (1995; Zbl 0843.58077)] and its references).
scientific article; zbMATH DE number 6815246
scientific article