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If the degree of $g$ is greater than the first dynamical degree of $f$, then the return sets of the system $(X\\times C,f\\times g)$ have the"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov multiplier of any iteration is not an integer. (Abstract, result (i))","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The paper depends on the standard functoriality and Northcott-type properties of height functions continuing to hold for endomorphisms of projective varieties over fields of arbitrary characteristic, allowing the derivation of the required height inequalities from the non-integer multiplier condition. 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