April  2014, 8(2): 221-250. doi: 10.3934/jmd.2014.8.221

Pseudo-automorphisms with no invariant foliation

1. 

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, United States

2. 

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

3. 

Department of Mathematics, Florida State University, Tallahassee, FL 32306, United States

Received  October 2013 Revised  May 2014 Published  November 2014

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.
Citation: Eric Bedford, Serge Cantat, Kyounghee Kim. Pseudo-automorphisms with no invariant foliation. Journal of Modern Dynamics, 2014, 8 (2) : 221-250. doi: 10.3934/jmd.2014.8.221
References:
[1]

T. Bayraktar and S. Cantat, Constraints on automorphism groups of higher dimensional manifolds, J. Math. Anal. Appl., 405 (2013), 209-213. doi: 10.1016/j.jmaa.2013.03.048.

[2]

E. Bedford, J. Diller and K. Kim, Pseudoautomorphisms with invariant elliptic curves, to appear in Proceedings of The Abel Symposium 2013, arXiv:1401.2386.

[3]

E. Bedford and K. Kim, Dynamics of (pseudo) automorphisms of 3-space: Periodicity versus positive entropy, Publ. Mat., 58 (2014), 65-119. doi: 10.5565/PUBLMAT_58114_04.

[4]

E. Bedford and K. Kim, Pseudo-automorphisms without dimension-reducing factors, Manuscript, 2012.

[5]

M.-J. Bertin and M. Pathiaux-Delefosse, Conjecture de Lehmer et petits nombres de Salem, Queen's Papers in Pure and Applied Mathematics, 81, Queen's University, Kingston, ON, 1989.

[6]

S. Cantat, Quelques aspects des systèmes dynamiques polynomiaux: Existence, exemples, rigidité, in Quelques aspects des systèmes dynamiques polynomiaux, Panor. Synthèses, 30, Soc. Math. France, Paris, 2010, 13-95.

[7]

S. Cantat, Dynamics of automorphisms of compact complex surfaces, in Frontiers in Complex Dynamics: In celebration of John Milnor's 80th birthday, Princeton Mathematical Series, Princeton University Press, Princeton, 2013, 463-514.

[8]

S. Cantat and C. Favre, Symétries birationnelles des surfaces feuilletées, J. Reine Angew. Math., 561 (2003), 199-235. doi: 10.1515/crll.2003.066.

[9]

G. Casale, Enveloppe galoisienne d'une application rationnelle de $\mathbbP^1$, Publ. Mat., 50 (2006), 191-202. doi: 10.5565/PUBLMAT_50106_10.

[10]

G. Casale, The Galois groupoid of Picard-Painlevé VI equation, in Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and their Deformations. Painlevé Hierarchies, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, 15-20.

[11]

J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math., 123 (2001), 1135-1169. doi: 10.1353/ajm.2001.0038.

[12]

T.-C. Dinh and N. Sibony, Une borne supérieure pour l'entropie topologique d'une application rationnelle, Ann. of Math. (2), 161 (2005), 1637-1644. doi: 10.4007/annals.2005.161.1637.

[13]

M. H. Gizatullin, Rational $G$-surfaces, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 110-144, 239. doi: 10.1070/IM1981v016n01ABEH001279.

[14]

M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2), 49 (2003), 217-235.

[15]

V. Guedj, Propriétés ergodiques des applications rationnelles, in Quelques Aspects des Systèmes Dynamiques Polynomiaux, Panor. Synthèses, 30, Soc. Math. France, Paris, 2010, 97-202.

[16]

S. R. Kaschner, R. A. Pérez and R. K. W. Roeder, Examples of rational maps of $\mathbb{CP}^2$ with equal dynamical degrees and no invariant foliation, preprint, arXiv:1309.4364, (2013), 11 pp.

[17]

B. Malgrange, On nonlinear differential Galois theory, Dedicated to the memory of Jacques-Louis Lions, Chinese Ann. Math. Ser. B, 23 (2002), 219-226. doi: 10.1142/S0252959902000213.

[18]

B. Malgrange, On nonlinear differential Galois theory, in Frontiers in Mathematical Analysis and Numerical Methods, World Sci. Publ., River Edge, NJ, 2004, 185-196.

[19]

B. Malgrange, Le groupoï de Galois d'un feuilletage, in Essays on Geometry and Related Topics, Vol. 1, 2, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001, 465-501.

[20]

K. Oguiso and T. T. Truong, Explicit examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy, preprint, arXiv:1306.1590, (2013), 1-12.

[21]

T. T. Truong, On automorphisms of blowups of $\mathbbP^3$, preprint, arXiv:1202.4224, (2012), 1-21.

[22]

A. P. Veselov, What is an integrable mapping?, in What is integrability?, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, 251-272. doi: 10.1007/978-3-642-88703-1_6.

[23]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.

[24]

D.-Q. Zhang, The $g$-periodic subvarieties for an automorphism $g$ of positive entropy on a compact Kähler manifold, Adv. Math., 223 (2010), 405-415. doi: 10.1016/j.aim.2009.08.010.

show all references

References:
[1]

T. Bayraktar and S. Cantat, Constraints on automorphism groups of higher dimensional manifolds, J. Math. Anal. Appl., 405 (2013), 209-213. doi: 10.1016/j.jmaa.2013.03.048.

[2]

E. Bedford, J. Diller and K. Kim, Pseudoautomorphisms with invariant elliptic curves, to appear in Proceedings of The Abel Symposium 2013, arXiv:1401.2386.

[3]

E. Bedford and K. Kim, Dynamics of (pseudo) automorphisms of 3-space: Periodicity versus positive entropy, Publ. Mat., 58 (2014), 65-119. doi: 10.5565/PUBLMAT_58114_04.

[4]

E. Bedford and K. Kim, Pseudo-automorphisms without dimension-reducing factors, Manuscript, 2012.

[5]

M.-J. Bertin and M. Pathiaux-Delefosse, Conjecture de Lehmer et petits nombres de Salem, Queen's Papers in Pure and Applied Mathematics, 81, Queen's University, Kingston, ON, 1989.

[6]

S. Cantat, Quelques aspects des systèmes dynamiques polynomiaux: Existence, exemples, rigidité, in Quelques aspects des systèmes dynamiques polynomiaux, Panor. Synthèses, 30, Soc. Math. France, Paris, 2010, 13-95.

[7]

S. Cantat, Dynamics of automorphisms of compact complex surfaces, in Frontiers in Complex Dynamics: In celebration of John Milnor's 80th birthday, Princeton Mathematical Series, Princeton University Press, Princeton, 2013, 463-514.

[8]

S. Cantat and C. Favre, Symétries birationnelles des surfaces feuilletées, J. Reine Angew. Math., 561 (2003), 199-235. doi: 10.1515/crll.2003.066.

[9]

G. Casale, Enveloppe galoisienne d'une application rationnelle de $\mathbbP^1$, Publ. Mat., 50 (2006), 191-202. doi: 10.5565/PUBLMAT_50106_10.

[10]

G. Casale, The Galois groupoid of Picard-Painlevé VI equation, in Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and their Deformations. Painlevé Hierarchies, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, 15-20.

[11]

J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math., 123 (2001), 1135-1169. doi: 10.1353/ajm.2001.0038.

[12]

T.-C. Dinh and N. Sibony, Une borne supérieure pour l'entropie topologique d'une application rationnelle, Ann. of Math. (2), 161 (2005), 1637-1644. doi: 10.4007/annals.2005.161.1637.

[13]

M. H. Gizatullin, Rational $G$-surfaces, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 110-144, 239. doi: 10.1070/IM1981v016n01ABEH001279.

[14]

M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2), 49 (2003), 217-235.

[15]

V. Guedj, Propriétés ergodiques des applications rationnelles, in Quelques Aspects des Systèmes Dynamiques Polynomiaux, Panor. Synthèses, 30, Soc. Math. France, Paris, 2010, 97-202.

[16]

S. R. Kaschner, R. A. Pérez and R. K. W. Roeder, Examples of rational maps of $\mathbb{CP}^2$ with equal dynamical degrees and no invariant foliation, preprint, arXiv:1309.4364, (2013), 11 pp.

[17]

B. Malgrange, On nonlinear differential Galois theory, Dedicated to the memory of Jacques-Louis Lions, Chinese Ann. Math. Ser. B, 23 (2002), 219-226. doi: 10.1142/S0252959902000213.

[18]

B. Malgrange, On nonlinear differential Galois theory, in Frontiers in Mathematical Analysis and Numerical Methods, World Sci. Publ., River Edge, NJ, 2004, 185-196.

[19]

B. Malgrange, Le groupoï de Galois d'un feuilletage, in Essays on Geometry and Related Topics, Vol. 1, 2, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001, 465-501.

[20]

K. Oguiso and T. T. Truong, Explicit examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy, preprint, arXiv:1306.1590, (2013), 1-12.

[21]

T. T. Truong, On automorphisms of blowups of $\mathbbP^3$, preprint, arXiv:1202.4224, (2012), 1-21.

[22]

A. P. Veselov, What is an integrable mapping?, in What is integrability?, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, 251-272. doi: 10.1007/978-3-642-88703-1_6.

[23]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.

[24]

D.-Q. Zhang, The $g$-periodic subvarieties for an automorphism $g$ of positive entropy on a compact Kähler manifold, Adv. Math., 223 (2010), 405-415. doi: 10.1016/j.aim.2009.08.010.

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