Itinerary lemma for some classes of discontinuous functions (Q2433623)

This is an abstract of the lecture during the XXVII Summer Symposium in Real Analysis, Opava 2003. The author presents a generalization of the itinerary lemma -- one of the basic tools in one-dimensional dynamic -- onto the class of all connected \(G_{\delta}\) functions from \([0,1]\) onto \([0,1]\); see the author [Real Anal. Exch. 29, No. 1, 205--209 (2003-2004; Zbl 1065.26004)]. Using this theorem the following results are obtained: (1) The composition of finitely many connected \(G_{\delta}\) functions from \([0,1]\) onto \([0,1]\) has a fixed point; see the reviewer [ibid. 29, No. 2, 931--938 (2003-2004; Zbl 1068.26004)]. (2) Sharkovskii's theorem holds for connected \(G_{\delta}\) functions; see the author [Fundam. Math. 179, No. 1, 27--41 (2003; Zbl 1070.26004)]. (3) There exists a \(DB_1\) function \(f\) from \([0,1]\) onto \([0,1]\) such that any non-empty closed set \(F\subset [0,1]\) can be translated to an \(\omega\)-limit set \(\tilde{F}\) for \(f\).