On the density of periodic measures for piecewise monotonic maps and their coding spaces

We prove that for all transitive piecewise monotonic maps, the following conditions are equivalent: (1) All ergodic measures on $[0,1]$ are approximated by periodic ones; (2) All invariant measures on coding spaces are approximated by periodic ones; (3) All ergodic measures on coding spaces are approximated by periodic ones. If we further assume that the map is piecewise increasing and either right or left continuous, then the following condition is also equivalent to (1)–(3). (4) All invariant measures on $[0,1]$ are approximated by periodic ones. We also construct an example of a piecewise decreasing and right continuous map which satisfies (1)–(3), but does not satisfy (4).