On the dynamical degrees of meromorphic maps preserving a fibration

TL;DR

This paper investigates how the dynamical degrees of a meromorphic map on a compact Kähler manifold relate to those of the induced map on the base of a preserved fibration, revealing new properties of such fibrations.

Contribution

It provides a formula linking the dynamical degrees of the map with those of the induced map on the base, advancing understanding of meromorphic maps preserving fibrations.

Findings

Dynamical degrees can be expressed in terms of relative degrees and base map degrees.

New properties of fibrations are derived from the dynamical behavior of the map.

Results apply to meromorphic maps on compact Kähler manifolds with preserved fibrations.

Abstract

Let f be a dominant meromorphic self-map on a compact Kaehler manifold X which preserves a fibration given by a meromorphic map from X to a compact Kaehler manifold Y. We compute the dynamical degrees of f in term of its dynamical degrees relative to the fibration and the dynamical degrees of the map g on Y which is induced by f. We derive from this result new properties of some fibrations intrinsically associated to X when this manifold admits an interesting dynamical system.