No semiconjugacy to a map of constant slope (Q2805063)

\textit{W. Parry} [Trans. Am. Math. Soc. 122, 368--378 (1966; Zbl 0146.18604)] showed that a continuous transitive piecewise monotone interval map is conjugate to an interval map of constant slope. Generalizing this result, \textit{J. Milnor} and \textit{W. Thurston} [Lect. Notes Math. 1342, 465--563 (1988; Zbl 0664.58015)] proved that a continuous piecewise monotone interval map is semiconjugate to an interval map of constant slope. In these papers of Parry [loc. cit.] and Milnor and Thurston [loc. cit.], the piecewise monotone interval maps have only a finite number of monotone subintervals.NEWLINENEWLINEMore recently, \textit{J. Bobok} [Stud. Math. 208, No. 3, 213--228 (2012; Zbl 1350.37043)] studied the class of continuous piecewise monotone Markov interval maps with countably many monotone subintervals. He gave a necessary and sufficient criteria for a map in the class to be semiconjugate to an interval map of constant slope.NEWLINENEWLINEIn the paper reviewed here, the authors study the class of countably piecewise continuous, piecewise monotone interval maps. They develop necessary and sufficient criteria for a map in the class to be semiconjugate to an interval map of constant slope. Also, sufficient conditions are given for when a map will not meet the criteria. Additionally, they provide examples of maps which do not meet the criteria and are not semiconjugate to interval maps of constant slope.