On the Higuchi fractal dimension of invariant measures for countable idempotent iterated function systems

The paper proves that for a γ-uniformly contractive max-plus countable IFS with normalized weights, M_{φ,q} is a Banach contraction on (I(X), d̃_{γ,τ}) with constant γ/τ and has a unique fixed point ν; moreover, the fixed points µ_n of the normalized finite partial systems converge to ν in the pointwise topology τ_p (Theorem 3.3 and Theorem 3.4). The contraction proof uses approximation by finite normalized systems and Theorem 3.2, then a summation-index shift in d̃_{γ,τ} to get γ/τ (, ). The candidate’s solution proves the same results via a clean single-scale estimate d_a(M_{φ,q}(µ), M_{φ,q}(ν)) ≤ d_{γ a}(µ,ν) and then summing to obtain the γ/τ contraction; it also establishes µ_n → ν in τ_p by a subsequence argument using limsup–sup interchanges. Both are correct; the proofs differ in technique. Definitions and set-up match the paper’s notions of ucCIFS/mpCIFS, the transfer operator M_{φ,q}, and τ_p/d̃_{β,τ} (, , ).