Hurwitz correspondences on compactifications of M0,N

Abstract

Hurwitz correspondences are certain multivalued self-maps of the moduli space M 0,N . They arise in the study of Thurston's topological characterization of rational functions. We consider the dynamics of Hurwitz correspondences and ask: On which compactifications of M 0,N should they be studied? We compare a Hurwitz correspondence H across various modular compactifications of M 0,N , and find a weighted stable curves compactification X † N that is optimal for its dynamics. We use X † N to show that the kth dynamical degree of H is the absolute value of the dominant eigenvalue of the pushforward induced by H on a natural quotient of H 2k (M 0,N ).

Dynamical degrees of Hurwitz correspondences

“…For any φ, the dynamical degrees of H φ are algebraic integers [Ram18]. As a parallel to Corollary 1.1, the results in [Ram18] show that, for k > 0, the degree over Q of k is 'likely' to decrease as k increases. More precisely, there is an upper bound for the degree over Q of k that decreases as k increases.…”

“…, where A full is a superset of A extending the functions F and rm. There is a finite covering map ν : H full → H, and we have the following commutative diagram (see [Ram18] for details).…”

“…More precisely, there is an upper bound for the degree over Q of k that decreases as k increases. In spite of the parallel, the methods used in this paper are very different from those used in [Ram18].…”

Dynamical invariants of monomial correspondences

“…Thus computing λ k involves computing infinitely many potentially unrelated pullback maps. Therefore, they have only been computed in low dimension or for maps which preserve certain geometric constraints: they are known for regular morphisms, for birational maps of surfaces [DF01], for endomorphisms of the affine plane [FJ11], for monomial maps [Lin12,FW12], for birational maps of hyperkähler varieties [LB17] and for Hurwitz correspondences (a class of mappings and correspondences obtained from Teichmüller theory in the work of Koch [Koc13]) [KR16,Ram18a,Ram18b]. The properties of dynamical degrees were studied for particular classes of birational transformations of P 3 [DH16, CD18, DL18, Dan18].…”

Degree growth of rational maps induced from algebraic structures

“…For monomial maps, the degree growth is obtained in [FW, L]. The degrees of special maps on the moduli space of marked points on the Riemann sphere have been investigated in [KR,R1,R2]. The dynamical degrees of maps preserving fibrations were computed in [DN].…”