Dynamical degrees of automorphisms on abelian varieties

The paper proves Theorem 2.1 via an explicit quaternion-algebra/PEL construction using a unit-circle conjugate γ of a given Salem number and fixed-point/norm formulas to read off analytic eigenvalues and hence dynamical degrees; see the statement of Theorem 2.1 and the construction manual with Propositions 2.3–2.4 and Theorem 2.5 for the divisional quaternion algebra, and the eigenvalue extraction via equations (1)–(2) in the proof of part (i) and its analogue in part (ii) . The candidate solution achieves the same dynamical-degree sequences by a different, though closely related, route: it embeds E=Q(λ) (resp. L=E⊗F K) into a suitable quaternion division algebra over F (resp. K), then uses Type II/IV PEL constructions and a linear-algebra lemma on H^{1,0} to compute λk from the moduli of analytic eigenvalues. The formula used for abelian varieties, that λk equals the product of the squares of the k largest moduli of analytic eigenvalues, matches the paper’s description that λk is the product of the largest 2k eigenvalues of the rational representation ρr(f)=ρa(f)⊕ρa(f) (hence the square) . Minor gaps in the model (order-embedding details and the identification of the (1,0)-eigenvalues at each archimedean place) are standard and addressable with cited number-theoretic/PEL facts; the paper’s proof supplies these steps explicitly via local-global and fixed-point norm identities.