Journal of Modern Dynamics (JMD)

Pseudo-automorphisms with no invariant foliation

Pages: 221 - 250, Issue 2, June 2014      doi:10.3934/jmd.2014.8.221

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Eric Bedford - Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, United States (email)
Serge Cantat - Département de Mathématiques et Applications (DMA), ENS Ulm, Paris, rue d’Ulm, France – Institut de Recherches Mathématiques de Rennes (IRMAR), Université de Rennes 1, UMR 6625 du CNRS, Bât. 22–23 du campus de Beaulieu, 35042 Rennes cedex, France (email)
Kyounghee Kim - Department of Mathematics, Florida State University, Tallahassee, FL 32306, United States (email)

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.

Keywords:  Pseudo-automorphism, invariant foliation, singular foliation, cohomologically nonhyperbolic, regular birational map, rational 3-folds, dynamical degrees.
Mathematics Subject Classification:  Primary: 37F10; Secondary: 32H50, 14E07.

Received: October 2013;      Revised: May 2014;      Available Online: November 2014.