On Slow Growth and Entropy-Type Invariants

Abstract

We discuss a generalization of topological entropy in which the usual exponential growth-rate function is replaced by an arbitrary gauge function. This generalized topological entropy had previously been described by Galatolo in 2003—up to a choice of notation in the defining formulas—which in turn is essentially the same as that described by Zhao and Pesin in 2015 (that involves a re-parameterization of time). One of the main motivations for studying this new set of invariants comes from the need to distinguish maps with zero (standard) topological entropy. In such cases, if the dynamics is not equicontinuous, then there exists at least one gauge for which the corresponding generalized entropy is positive. After illustrating this simple qualitative criterion, we perform a more quantitative study of the growth of orbits in some low-dimensional examples of zero-entropy maps. Our examples include period-doubling maps in dimension one, and maps of the annulus built from circle homeomorphisms having an exceptional minimal set.