Irregular Sets Carry Full Metric Mean Dimension

The paper proves that for systems with the specification property, if the irregular set Iϕ is nonempty then it carries full Bowen upper and lower metric mean dimension equal to the classical metric mean dimension of the whole system (Theorem A) , via a Moran-like construction that alternates long orbit segments biased toward two invariant measures with distinct ϕ-averages and an entropy-distribution argument to lower-bound hBtop at a fixed scale, then passes to metric mean dimension using variational principles for partitions and Katok-type formulas . The candidate solution gives a different (combinatorial) construction: a specification-based ‘horseshoe with markers’ built from a large library of (ℓ,ε)-separated blocks with nearly constant ϕ-averages, plus alternating marker blocks enforcing nonconvergence of Birkhoff averages, and then estimates Bowen entropy per scale ε to match htop(f,X,ε) up to a vanishing overhead, which yields the same mean-dimension conclusion. The paper’s proof is measure-theoretic and uses entropy of partitions with a Moran fractal and an entropy distribution principle (e.g., Lemmas 3.6–3.8 and the final bound chain) , while the model’s proof is a direct separated-set/gluing argument. Both are correct; they reach the same result by different routes.