Positive subexponential entropy for universal Turing machines

Ascertain whether, for a given universal Turing machine, there exists a subexponential growth entropy invariant—such as polynomial entropy—that is positive, even in cases where the topological entropy may vanish.

Background

The authors note that while many universal Turing machines (in a regular class they studied) have positive topological entropy, some universal Turing machines can have zero topological entropy. They point to refined entropy notions of subexponential growth (e.g., polynomial entropy) that can detect complexity beyond topological entropy.

The problem asks whether every universal Turing machine admits some positive subexponential entropy invariant, thereby quantifying computational complexity in systems with zero topological entropy.