Density of periodic orbit measures for transformations on the interval with two monotonic pieces

Transformations T : [0, 1] → [0, 1] with two monotonic pieces are considered. Under the assumption that T is topologically transitive and htop(T ) > 0, it is proved that the invariant measures concentrated on periodic orbits are dense in the set of all invariant probability measures. Introduction. In order to investigate generic properties of invariant measures for a topological dynamical system R. Bowen [2] introduced the specification property. This is a topological property which implies that the measures concentrated on periodic orbits are dense in the set of all invariant measures. The specification property implies generic properties for different types of invariant measures, e.g. ergodic measures, nonatomic measures, measures with zero entropy and strongly mixing measures (see [3]). It is known that the specification property holds for basic sets of axiom A-diffeomorphisms ([2], [3]), for monotonic mod one transformations ([5]) and for continuous maps on the interval ([1]). We investigate in this paper dynamical systems generated by piecewise monotonic maps. If these maps have discontinuities, it becomes complicated to prove the density of periodic orbit measures. Besides generic properties of invariant measures there are two more reasons to consider this problem for piecewise monotonic maps T : [0, 1]→ [0, 1]. We describe these reasons below. 1991 Mathematics Subject Classification: 58F03, 58F11, 54H20.