Rigidity and flexibility of polynomial entropy

We introduce the notion of a one-way horseshoe for topological dynamical systems and show that, quite surprisingly, it plays the same role in the theory of polynomial entropy as the notion of a horseshoe plays in the theory of topological entropy. Indeed, we show that the existence of a one-way horseshoe gives a lower bound for polynomial entropy and for maps of the interval also conversely, polynomial entropy is given by one-way horseshoes of iterates of the map, analogously to Misiurewicz's theorem on topological entropy and standard ‘two-way’ horseshoes. As a consequence we get a rigidity result that if the polynomial entropy of an interval map is finite, then it is an integer. Furthermore, for interval maps of Sharkovskii type 1 the polynomial entropy can also be computed by what we call chains of essential intervals. We further describe the possible values of polynomial entropy of maps of all Sharkovskii types. We then apply our tools to the classic logistic family, showing that the polynomial entropy increases weakly monotonically with the parameter, making discrete jumps up at bifurcation points along the period-doubling cascade. On the other hand, we show that in the class of all continua the polynomial entropy of continuous maps is very flexible. For every value α∈[0,∞] there is a homeomorphism on a continuum with polynomial entropy α. We discuss also possible values of the polynomial entropy of continuous maps on dendrites.