Pseudo-automorphisms with no invariant foliation

TL;DR

This paper constructs a specific birational transformation of a rational threefold with equal dynamical degrees greater than one, which does not preserve any invariant foliation, answering a previously open question.

Contribution

It provides the first example of such a transformation, developing new techniques to analyze invariant foliations under birational maps.

Findings

First example of a birational transformation with equal dynamical degrees >1 and no invariant foliation

Developed new methods to study foliations invariant under birational transformations

Negatively answers a question posed by Guedj

Abstract

We construct an example of a birational transformation of a rational threefold for which the first and second dynamical degrees coincide and are $>1$, but which does not preserve any holomorphic (singular) foliation. In particular, this provides a negative answer to a question of Guedj. On our way, we develop several techniques to study foliations which are invariant under birational transformations.