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On the singular-hyperbolicity of star flows
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 |
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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