The Forbidden Swap

The classical variational principle in dynamical systems says: the topological entropy of a map equals the supremum of measure-theoretic entropies over all invariant measures. It’s one of the cleanest bridges between topology and measure theory — the worst-case measure captures the full complexity of the dynamics.

Carvalho and Pessil investigate whether the same principle holds for metric mean dimension, a finer invariant that captures complexity below the scale where entropy diverges. They introduce “mean quantization dimension” for a measure and prove the upper metric mean dimension equals the maximum quantization dimension over invariant measures.

But the variational principle’s clean form requires swapping two limits: the supremum over measures and the limit as the resolution scale approaches zero. The central result is negative — for several well-known entropy functions in the literature, this swap is not valid. The order matters: taking the supremum first gives a different answer from taking the limit first. These entropy functions fail the variational principle not because of any defect in the dynamics or the measures, but because the interchange of limit operations is illegitimate.

The structural point: the variational principle is not a theorem about dynamics. It is a theorem about when limits commute. When they do, topology and measure theory agree. When they don’t, you get a gap — a topological complexity that no single invariant measure can witness. The dynamical content is real, but the mathematical obstruction is analytic: the order in which you zoom in and optimize determines whether the bridge exists.