American Institute of Mathematical Sciences

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

Pseudo-automorphisms with no invariant foliation

Eric Bedford 1, , Serge Cantat 2, and  Kyounghee Kim 3,

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.
Keywords: rational 3-folds, cohomologically nonhyperbolic, dynamical degrees., invariant foliation, regular birational map, singular foliation, Pseudo-automorphism.
Mathematics Subject Classification: Primary: 37F10; Secondary: 32H50, 14E0.
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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[1]

Yong Fang, Patrick Foulon, Boris Hasselblatt. Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows. Electronic Research Announcements, 2010, 17: 80-89. doi: 10.3934/era.2010.17.80
[2]

M. Jotz. The leaf space of a multiplicative foliation. Journal of Geometric Mechanics, 2012, 4 (3) : 313-332. doi: 10.3934/jgm.2012.4.313
[3]

Jeffrey J. Early, Juha Pohjanpelto, Roger M. Samelson. Group foliation of equations in geophysical fluid dynamics. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1571-1586. doi: 10.3934/dcds.2010.27.1571
[4]

José Santana Campos Costa, Fernando Micena. Pathological center foliation with dimension greater than one. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1049-1070. doi: 10.3934/dcds.2019044
[5]

Doris Bohnet. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. Journal of Modern Dynamics, 2013, 7 (4) : 565-604. doi: 10.3934/jmd.2013.7.565
[6]

Connor Mooney, Yang Yang. A proof by foliation that lawson's cones are $ A_{\Phi} $-minimizing. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021077
[7]

Jeffrey Diller, Han Liu, Roland K. W. Roeder. Typical dynamics of plane rational maps with equal degrees. Journal of Modern Dynamics, 2016, 10: 353-377. doi: 10.3934/jmd.2016.10.353
[8]

Sze-Bi Hsu, Ming-Chia Li, Weishi Liu, Mikhail Malkin. Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1465-1492. doi: 10.3934/dcds.2003.9.1465
[9]

C. Chandre. Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 457-465. doi: 10.3934/dcdsb.2002.2.457
[10]

Mazyar Ghani Varzaneh, Sebastian Riedel. A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4587-4612. doi: 10.3934/dcdsb.2020304
[11]

Piotr Pokora, Tomasz Szemberg. Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone. Electronic Research Announcements, 2014, 21: 126-131. doi: 10.3934/era.2014.21.126
[12]

Dino Festi, Alice Garbagnati, Bert Van Geemen, Ronald Van Luijk. The Cayley-Oguiso automorphism of positive entropy on a K3 surface. Journal of Modern Dynamics, 2013, 7 (1) : 75-97. doi: 10.3934/jmd.2013.7.75
[13]

Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433
[14]

Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639
[15]

Marie-Claude Arnaud. A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map. Journal of Modern Dynamics, 2011, 5 (3) : 583-591. doi: 10.3934/jmd.2011.5.583
[16]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089
[17]

Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002
[18]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285
[19]

Alexey A. Petrov, Sergei Yu. Pilyugin. Shadowing near nonhyperbolic fixed points. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3761-3772. doi: 10.3934/dcds.2014.34.3761
[20]

Hieu Trung Do, Thomas A. Schmidt. New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant. Journal of Modern Dynamics, 2016, 10: 541-561. doi: 10.3934/jmd.2016.10.541

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences