Examples of rational maps of $CP^2$ with equal dynamical degrees and no invariant foliation

Abstract

Abstract. We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation.

Typical dynamics of plane rational maps with equal degrees

“…Section 2 gives the proofs of Theorems A and B. The main techniques that we use are hyperbolic dynamics, elementary geometry of algebraic curves, and an application of Theorem 4.1' from [30]. One can obtain some (though not nearly all) of the conclusions of Theorem A with less exertion by appealing to work of De Thélin and Vigny [8,Theorem 1].…”

“…It was demonstrated recently [30] that λ 1 ( f ) = λ 2 ( f ) does not imply f preserves a rational fibration, nor indeed even a singular holomorphic foliation. Specifically, the following criterion appears in that paper: THEOREM 2.4 (Theorem 4.1' from [30]).…”

“…It was demonstrated recently [30] that λ 1 ( f ) = λ 2 ( f ) does not imply f preserves a rational fibration, nor indeed even a singular holomorphic foliation. Specifically, the following criterion appears in that paper: THEOREM 2.4 (Theorem 4.1' from [30]). Assume that f has an indeterminate point q such that 1. there are irreducible curves C n ⊂ f n (q) with lim sup deg(C n ) = ∞; and 2. q has an infinite preorbit along which f is a finite holomorphic map.…”

“…We refer the reader to [18,30] and the references therein for more details about the transformation of singular holomorphic foliations by rational maps.…”

“…Based on this evidence, Guedj asked whether any rational map with equal (maximal) dynamical degrees must preserve a fibration [26, p.103]. However, examples were recently found by Bedford-Cantat-Kim [3] and Kaschner-Pérez-Roeder [30] showing that this is not the case.…”