Entropies and spectra
This is a broad survey of various results concerning the asymptotic growth of ``certain quantities'' related to entropy of a given dynamical system. For example, when the dynamical system is defined by an integer (or real) action, the asymptotic growth rate of the number of periodic points (or periodic orbits) of (primitive) period \(p\) is related to the entropy if the underlying dynamics is ``hyperbolic''. For dynamical systems defined by the actions of discrete subgroups of isometries of the hyperbolic space, asymptotics of the growth rates of several analogous quantities have been studied. Yet another situation where similar notions and results can be developed is the example of group actions on graphs and trees. This paper describes various known results and it gives exact expressions for asymptotics of growth rates of various relevant quantities in all these situations.
scientific article; zbMATH DE number 720165