  We introduce the notion of a one-way horseshoe and show that the polynomial
entropy of an interval map is given by one-way horseshoes of iterates of the
map, obtaining in such a way an analogue of Misiurewicz's theorem on
topological entropy and standard `two-way' horseshoes. Moreover, if the map is
of Sharkovskii type 1 then its polynomial entropy can also be computed by what
we call chains of essential intervals. As a consequence we get a rigidity
result that if the polynomial entropy of an interval map is finite, then it is
an integer. We also describe the possible values of polynomial entropy of maps
of all Sharkovskii types. As an application we compute the polynomial entropy
of all maps in the logistic family. 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 $\alpha \in [0,\infty]$ there is a homeomorphism on a
continuum with polynomial entropy $\alpha$. We discuss also possible values of
the polynomial entropy of continuous maps on dendrites.

Dynamical Systems