Weighted Topological Entropy of Random Dynamical Systems

The paper proves a weighted variational principle for random dynamical systems (Theorem 1.2) using an a-weighted Bowen entropy and fiber entropies on a factor, together with a lower bound via a weighted Brin–Katok formula and an upper bound via a dynamical Frostman lemma and entropy-averaging lemmas. The candidate solution follows the same two-sided scheme: (i) lower bound by relating a-weighted Bowen balls to atoms of a joined partition and applying conditional SMB; (ii) upper bound by a Carathéodory/Frostman measure with exponential ball bounds, plus averaging to an invariant measure and conditional entropy estimates. The small differences are technical: the paper constructs measures for single base points ω0 and then averages (easing measurability issues), while the model posits a random Frostman measure family {m_ω}. Overall the logical flow and core tools align closely with the paper’s argument, and no substantive contradiction was found (Theorem 1.2 statement and setup, definition of a-weighted Bowen entropy, lower bound via Brin–Katok, and the upper bound’s Frostman-entropy averaging steps all match) .