We consider a class of piecewise monotonically increasing functions f on the unit interval I. We want to determine the measures with maximal entropy for these transformations. In part I we construct a shift-space Σ f + isomorphic to (I, f) generalizing the -shift and another shift Σ M over an infinite alphabet, which is of finite type given by an infinite transition matrix M. Σ M has the same set of maximal measures as (I, f) and we are able to compute the maximal measures of maximal measures of. In part II we try to bring these results back to (I, f). There are only finitely many ergodic maximal measures for (I, f). The supports of two of them have at most finitely many points in common. If (I, f) is topologically transitive it has unique maximal measure. © 1979 Hebrew University.
