Dynamical degrees of Hurwitz correspondences

Abstract

Let $\unicode[STIX]{x1D719}$ be a post-critically finite branched covering of a two-sphere. By work of Koch, the Thurston pullback map induced by $\unicode[STIX]{x1D719}$ on Teichmüller space descends to a multivalued self-map—a Hurwitz correspondence ${\mathcal{H}}_{\unicode[STIX]{x1D719}}$—of the moduli space ${\mathcal{M}}_{0,\mathbf{P}}$. We study the dynamics of Hurwitz correspondences via numerical invariants called dynamical degrees. We show that the sequence of dynamical degrees of ${\mathcal{H}}_{\unicode[STIX]{x1D719}}$ is always non-increasing and that the behavior of this sequence is constrained by the behavior of $\unicode[STIX]{x1D719}$ at and near points of its post-critical set.

Hurwitz correspondences on compactifications of M0,N

“…In contrast, the action on the subspace Λ <(k) P is an artifact of the compactification M 0,P -this action is "contained in the boundary" (Corollary 10.8). We note that Theorems 9.6 and 10.6 do not help give an upper bound for the entropy of H. In ( [25]), we show that for any Hurwitz correspondence H, the sequence k → (kth dynamical degree of H) is nonincreasing. Thus the largest dynamical degree -the one providing an upper bound for entropy -is the 0th.…”

“…Theorem 4.6 as stated does not appear in the references quoted above. The proofs in [31] were modified in the Appendix to [25] to give a complete proof. Thus we can study the dynamical degrees of Γ : X X via the action of Γ on the birational model X ′ .…”

“…In this work, we exploit the moduli space interpretation of Hurwitz correspondences to provide an account of their homological actions on various compactifications of M 0,P . Together with a companion paper [25], this comprises the first systematic study of the dynamical degrees of any natural family of rational multivalued maps.…”

“…Definition 5.6 (Hurwitz correspondence,[25],Definition 2.19). Let A ′ be any subset of A with cardinality at least 3.…”

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].…”