2016, 10: 23-32. doi: 10.3934/jmd.2016.10.23

Jonquières maps and $SL(2;\mathbb{C})$-cocycles

1. 

Institut de Mathématiques de Jussieu- Paris Rive Gauche, UMR 7586, Université Paris Diderot, Bâtiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France

Received  April 2014 Published  February 2016

We started the study of the family of birational maps $(f_{\alpha,\beta})$ of $\mathbb{P}^2_\mathbb{C}$ in [12]. For ``$(\alpha,\beta)$ well chosen'' of modulus $1$, the centraliser of $f_{\alpha,\beta}$ is trivial, the topological entropy of $f_{\alpha,\beta}$ is $0$, and there exist two domains of linearisation: in the first one the closure of the orbit of a point is a torus, in the other one the closure of the orbit of a point is the union of two circles. On $\mathbb{P}^1_\mathbb{C}\times \mathbb{P}^1_\mathbb{C}$, any $f_{\alpha,\beta}$ can be viewed as a cocyle; using recent results about $\mathrm{SL}(2;\mathbb{C})$-cocycles ([1]), we determine the Lyapunov exponent of the cocyle associated to $f_{\alpha,\beta}$.
Citation: Julie Déserti. Jonquières maps and $SL(2;\mathbb{C})$-cocycles. Journal of Modern Dynamics, 2016, 10: 23-32. doi: 10.3934/jmd.2016.10.23
References:
[1]

A. Avila, Global theory of one-frequency Schrödinger operators,, Acta Math., 215 (2015), 1. doi: 10.1007/s11511-015-0128-7. Google Scholar

[2]

A. Avila, S. Jitomirskaya and C. Sadel, Complex one-frequency cocycles,, J. Eur. Math. Soc. (JEMS), 16 (2014), 1915. doi: 10.4171/JEMS/479. Google Scholar

[3]

A. Beauville, Complex Algebraic Surfaces,, Translated from the 1978 French original by R. Barlow, (1978). doi: 10.1017/CBO9780511623936. Google Scholar

[4]

E. Bedford and K. Kim, Periodicities in linear fractional recurrences: Degree growth of birational surface maps,, Michigan Math. J., 54 (2006), 647. doi: 10.1307/mmj/1163789919. Google Scholar

[5]

E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Linear fractional recurrences,, J. Geom. Anal., 19 (2009), 553. doi: 10.1007/s12220-009-9077-8. Google Scholar

[6]

E. Bedford and K. Kim, Continuous families of rational surface automorphisms with positive entropy,, Math. Ann., 348 (2010), 667. doi: 10.1007/s00208-010-0498-2. Google Scholar

[7]

E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Rotations domains,, Amer. J. Math., 134 (2012), 379. doi: 10.1353/ajm.2012.0015. Google Scholar

[8]

J. Blanc and J. Déserti, Degree growth of birational maps of the plane,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14 (2015), 507. Google Scholar

[9]

S. Cantat, Dynamique des automorphismes des surfaces projectives complexes,, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 901. doi: 10.1016/S0764-4442(99)80294-8. Google Scholar

[10]

S. Cantat, Sur les groupes de transformations birationnelles des surfaces,, Ann. of Math. (2), 174 (2011), 299. doi: 10.4007/annals.2011.174.1.8. Google Scholar

[11]

D. Cerveau and J. Déserti, Centralisateurs dans le groupe de Jonquières,, Michigan Math. J., 61 (2012), 763. doi: 10.1307/mmj/1353098512. Google Scholar

[12]

J. Déserti, Expériences sur certaines transformations birationnelles quadratiques,, Nonlinearity, 21 (2008), 1367. doi: 10.1088/0951-7715/21/6/013. Google Scholar

[13]

J. Déserti and J. Grivaux, Automorphisms of rational surfaces with positive entropy,, Indiana Univ. Math. J., 60 (2011), 1589. doi: 10.1512/iumj.2011.60.4427. Google Scholar

[14]

J. Diller, Cremona transformations, surface automorphisms, and plane cubics,, With an appendix by I. Dolgachev, 60 (2011), 409. doi: 10.1307/mmj/1310667983. Google Scholar

[15]

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

[16]

M. H. Gizatullin, Rational $G$-surfaces,, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 110. Google Scholar

[17]

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

[18]

C. T. McMullen, Dynamics on blowups of the projective plane,, Publ. Math. Inst. Hautes Études Sci., 105 (2007), 49. doi: 10.1007/s10240-007-0004-x. Google Scholar

[19]

H. Rüssmann, Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition,, Ergodic Theory Dynam. Systems, 22 (2002), 1551. doi: 10.1017/S0143385702000974. Google Scholar

[20]

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

show all references

References:
[1]

A. Avila, Global theory of one-frequency Schrödinger operators,, Acta Math., 215 (2015), 1. doi: 10.1007/s11511-015-0128-7. Google Scholar

[2]

A. Avila, S. Jitomirskaya and C. Sadel, Complex one-frequency cocycles,, J. Eur. Math. Soc. (JEMS), 16 (2014), 1915. doi: 10.4171/JEMS/479. Google Scholar

[3]

A. Beauville, Complex Algebraic Surfaces,, Translated from the 1978 French original by R. Barlow, (1978). doi: 10.1017/CBO9780511623936. Google Scholar

[4]

E. Bedford and K. Kim, Periodicities in linear fractional recurrences: Degree growth of birational surface maps,, Michigan Math. J., 54 (2006), 647. doi: 10.1307/mmj/1163789919. Google Scholar

[5]

E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Linear fractional recurrences,, J. Geom. Anal., 19 (2009), 553. doi: 10.1007/s12220-009-9077-8. Google Scholar

[6]

E. Bedford and K. Kim, Continuous families of rational surface automorphisms with positive entropy,, Math. Ann., 348 (2010), 667. doi: 10.1007/s00208-010-0498-2. Google Scholar

[7]

E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Rotations domains,, Amer. J. Math., 134 (2012), 379. doi: 10.1353/ajm.2012.0015. Google Scholar

[8]

J. Blanc and J. Déserti, Degree growth of birational maps of the plane,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14 (2015), 507. Google Scholar

[9]

S. Cantat, Dynamique des automorphismes des surfaces projectives complexes,, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 901. doi: 10.1016/S0764-4442(99)80294-8. Google Scholar

[10]

S. Cantat, Sur les groupes de transformations birationnelles des surfaces,, Ann. of Math. (2), 174 (2011), 299. doi: 10.4007/annals.2011.174.1.8. Google Scholar

[11]

D. Cerveau and J. Déserti, Centralisateurs dans le groupe de Jonquières,, Michigan Math. J., 61 (2012), 763. doi: 10.1307/mmj/1353098512. Google Scholar

[12]

J. Déserti, Expériences sur certaines transformations birationnelles quadratiques,, Nonlinearity, 21 (2008), 1367. doi: 10.1088/0951-7715/21/6/013. Google Scholar

[13]

J. Déserti and J. Grivaux, Automorphisms of rational surfaces with positive entropy,, Indiana Univ. Math. J., 60 (2011), 1589. doi: 10.1512/iumj.2011.60.4427. Google Scholar

[14]

J. Diller, Cremona transformations, surface automorphisms, and plane cubics,, With an appendix by I. Dolgachev, 60 (2011), 409. doi: 10.1307/mmj/1310667983. Google Scholar

[15]

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

[16]

M. H. Gizatullin, Rational $G$-surfaces,, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 110. Google Scholar

[17]

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

[18]

C. T. McMullen, Dynamics on blowups of the projective plane,, Publ. Math. Inst. Hautes Études Sci., 105 (2007), 49. doi: 10.1007/s10240-007-0004-x. Google Scholar

[19]

H. Rüssmann, Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition,, Ergodic Theory Dynam. Systems, 22 (2002), 1551. doi: 10.1017/S0143385702000974. Google Scholar

[20]

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

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