Marked length spectrum rigidity for Anosov magnetic surfaces

The paper proves that on a closed surface, equality of the magnetic marked length spectra together with equal area and cohomology class forces an isometry and equality of magnetic intensities up to a diffeomorphism isotopic to the identity (Theorem 1.3). The argument proceeds via MLS ⇒ C0 orbit conjugacy (Livšic+Ghys), area ⇒ volume-preserving, then smooth conjugacy by Gogolev–Rodriguez Hertz, reduction to the conformal case by Echevarría Cuesta, and a Katok-style Jensen argument plus homological fullness to kill the conformal factor and identify the magnetic intensity (Theorems 1.3/1.4; Lemma 2.1; Theorem 3.1). The candidate solution outlines precisely this blueprint. The only substantive slip is a misidentification of the Liouville volume form (the paper uses μg = −αg ∧ dαg), but this does not affect the main line of reasoning given the area-based volume-preservation step (Lemma 2.1) and subsequent rigidity reductions. Overall, both are correct and follow substantially the same proof strategy, with the paper supplying rigorous lemmas for the homological fullness/exactness and the conformal reduction steps (e.g., Theorem 3.1 and Lemma 3.2).