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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[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.

[20]

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

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[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.

[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.

[20]

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

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