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Abstract:
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In this thesis, we applied a method for computing the polynomial entropy of
dynamical systems with a finite non-wandering set to certain hyperspaces of the interval and
the circle. We subsequently generalized these results to the case where the non-wandering
set is not finite. We proved that the polynomial entropy of the induced map C(f ) is equal
to 2, that of Fn(f ) is equal to n, and that of 2f is infinite, when f is a homeomorphism of
the interval or the circle. Furthermore, we generalized these results for the n-fold symmetric
product of the interval and the circle to an arbitrary compact metric space X and extended
them to the suspension of the space Fn(X). In addition to the results on hyperspaces, we
presented results concerning polynomial entropy on certain one-dimensional continua, such
as local dendrites and regular curves. We also constructed a family of pointwise periodic
homeomorphisms on a continuum whose polynomial entropy attains every value in [0, +∞]. |