Polynomial entropy and expansivity

Abstract

In this paper we study the polynomial entropy of homeomorphism on compact metric space. We construct a homeomorphism on a compact metric space with vanishing polynomial entropy that it is not equicontinuous. Also we give examples with arbitrarily small polynomial entropy. Finally, we show that expansive homeomorphisms and positively expansive maps of compact metric spaces with infinitely many points have polynomial entropy greater or equal than 1.

Rigidity and Flexibility of Polynomial Entropy

“…On the flexibility side, in Theorem 37 we show that for homeomorphisms on continua, polynomial entropy can take arbitrary values in [0, ∞]. This already gives a complete positive answer to the above-mentioned Problem 3 from [ACM17] and, similarly as in [HL19], a strong negative answer to Problem 4.…”

“…In the present paper we do not add more dynamical assumptions on maps, such as expansivity, equicontinuity, distality, or any kind of smoothness and the like (for some results on polynomial entropy under additional assumptions on the map see [L13], [ACM17], [Mar13], and a recent paper [CPR21] where a question from [ACM17] is answered negatively).…”

“…• The set of values of the polynomial entropies of homeomorphisms of compact metric spaces is dense in (0, ∞) [ACM17, Remark 3.6]. Moreover, in [ACM17] the authors asked the following questions.…”

“…• (Problem 3 in [ACM17]) Is every positive real number the polynomial entropy of a homeomorphism of a compact metric space? • (Problem 4 in [ACM17]) If f is a homeomorphism of a connected compact metric space with h pol (f ) finite, is it true that h pol (f ) is an integer number? In [HL19] the authors, apparently not aware of [ACM17], proved the following.…”

“…• (Problem 4 in [ACM17]) If f is a homeomorphism of a connected compact metric space with h pol (f ) finite, is it true that h pol (f ) is an integer number? In [HL19] the authors, apparently not aware of [ACM17], proved the following.…”