GENERALIZED q-DIMENSIONS OF MEASURES ON NONAUTONOMOUS FRACTALS

The paper proves sharp upper and lower bounds D_q(μ^ω) ≤ min{d_q^+, d} and, under i.i.d. absolutely continuous random translations, D_q(μ^ω) ≥ min{d_q^−, d} for q>1, via stopping-time covers and multienergy (join-tree) estimates, respectively, and then identifies d_q^− = d_q^+ = d_q in the almost self-affine (stationary) case, yielding D_q(μ^ω) = min{d, d_q} for q ≥ 1. These are stated as Theorem 2.8 (upper bound) and Theorem 2.9 (lower bound) with Proposition 2.10 and Corollary 2.11 giving existence and the final formula; the q=1 identification is handled by showing d_1 = d_1^− = d_1^+ and hence D_1(μ^ω) = min{d_1, d} (see the derivation culminating in inequality (5.21)). All of these steps appear explicitly in the paper’s statements and proofs, e.g., the definitions of d_q^± via the stopping-time Σ^*(s,r) sums and the multienergy kernel bounds used to control E[∫ μ^ω(B(x,r))^{q−1} dμ^ω(x)] (Theorem 2.8, Theorem 2.9, Corollary 2.11, and the multienergy Proposition 5.2) . The candidate solution follows the same structure: (i) a stopping-time/singular-value upper bound, (ii) a multienergy-based probabilistic lower bound summed over join trees, and (iii) a q=1 argument leading to D_1 = min{d, d_1}. Its stationary pressure definition of d_q via lim_k (∑_{|u|=k} ψ_s(T_u)^{1−q} p_u^q)^{1/k} = 1 matches the paper’s (2.30)–(2.31), and its conclusion that d_q^− = d_q^+ = d_q in the stationary case coincides with Corollary 2.11. Consequently, both are correct and use substantially the same proof strategy, with only minor presentational differences (e.g., the model invokes exact dimensionality for q=1, whereas the paper proves d_1^− = d_1^+ = d_1 directly) .