Topological entropy of piecewise embedding maps on regular curves

It is well known that in the dynamics of a piecewise strictly monotone (that is,piecewise embedding) map f on an interval, the topological entropy can be expressed interms of the growth of the number (that is, the lap number) of strictly monotone intervalsfor f n. Recently, there has been an increase in the importance of fractal sets in the sciences,and many geometric and dynamical properties of fractal sets have been studied. In thepresent paper, we shall study topological entropy of some maps on regular curves, whichare contained in the class of fractal sets. We generalize the theorem ofMisiurewicz–Szlenkand Young to the cases of regular curves and dendrites.