Let $(X,f)$ be a dynamical system with the specification property and φ be a continuous function. In this paper, we consider the multifractal irregular set {align*} Iφ=\{x∈ X:limn→∞1/n∑ᵢ₌₀ⁿ⁻¹φ(f^ix)\ does not exist\} {align*} and show that this set is either empty or carries full Bowen upper and lower metric mean dimension with potential.
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Zhang et al. (2024) studied this question.
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