HORSESHOES AND ENTROPY FOR PIECEWISE CONTINUOUS PIECEWISE MONOTONE MAPS

We show that for piecewise continuous piecewise monotone interval maps to define topological entropy one can use the standard Bowen's method, as well as the ideas of symbolic dynamics. As for continuous interval maps, positive entropy is due to the existence of horseshoes. This allows us to prove lower semi-continuity of topological entropy as a function of a map on some natural spaces of piecewise continuous piecewise monotone maps.