Hereditary lowerability of diagonal entropy scales

Abstract

Let $T$ be a continuous self-map of a compact metric space. For each integer $p\geq2$, we consider the upper capacity entropy scale obtained by dividing the $p$-fold logarithm of an orbit covering number by the $(p-1)$-fold logarithm of time. We prove that every nonempty compact set contains a compact subset realizing any value between zero and its own scale entropy. Every strictly intermediate value is realized by a countable compact set with exactly one accumulation point. The proof combines localization at a fixed spatial resolution, a finite-tree selection lemma, and refinements of orbit covers at dyadic times. An example shows that the assertion about one accumulation point does not extend to the upper endpoint. On the product-diagonal construction used in classical entropy lowerability, the diagonal scales preserve the values of the base subsets, although classical upper capacity entropy takes only zero or infinity.