Multifractal analysis of the Lyapunov exponent for random graph directed Markov systems

The paper proves, under normality, RBSC over finite subedges, and cofinitely regularity, that for a full-measure set of base points ω the Lyapunov level sets satisfy dim_H L_ω(β) = (1/β) inf_s (p(s)+βs) for β in (−p′(+∞), −p′(s_∞)), and are empty outside [−p′(+∞), −p′(s_∞)] (Theorem 1.1). It achieves this via finite-alphabet approximations, a two-parameter pressure, and fiberwise multifractal measures, yielding upper and lower bounds that match (Proposition 4.11, Proposition 4.10, Theorem 4.7, Lemma 4.12). The candidate solution reaches the same conclusion via a standard multifractal formalism: cylinder covers + pressure for the upper bound and random Gibbs/equilibrium measures on finite subsystems for the lower bound, then exhaustion. While the structures and required hypotheses align closely with the paper, the techniques differ in emphasis: the paper builds fiberwise multifractal measures to ensure uniform full-measure sets in s, whereas the model appeals to random RPF/Gibbs arguments and a countable intersection argument over rational s. Both are correct and consistent with each other, but not identical in proof strategy.