Intermediate topological entropies for subsets of nonautonomous dynamical systems

Abstract

Motivated by the notion of intermediate dimensions introduced by Falconer et al., we introduce a continuum of topological entropies that are intermediate between the (Bowen) topological entropy and the lower and upper capacity topological entropies. This is achieved by restricting the families of allowable covers in the definition of topological entropy by requiring that the lengths of all strings used in a particular cover satisfy $ N \le n < N/\theta + 1$, where $ \theta \in [0,1]$ is a parameter. When $ \theta = 1$, only covers using strings of the same length are allowed, and we recover the lower and upper capacity topological entropies; when $ \theta = 0$, there are no restrictions, and the definition coincides with the topological entropy. We first establish a quantitative inequality for the upper and lower intermediate topological entropies, which mirrors the corresponding result for intermediate dimensions. As a consequence, the intermediate topological entropies are continuous on $(0,1]$, though discontinuity may arise at $0$; an illustrative example is provided to demonstrate this phenomenon. We then investigate several fundamental properties of the intermediate topological entropies for nonautonomous dynamical systems, including the power rule, monotonicity and product formulas. Finally, we derive an inequality relating intermediate entropies with respect to factor maps.