Journal of Modern Dynamics (JMD)

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

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

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

Abstract: 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 $f_{\alpha,\beta}$.

Keywords:  $\mathbb{SL}(2;\mathbb{C})$-cocycle, Lyapunov exponent, birational map, Jonquières map.
Mathematics Subject Classification:  Primary: 37F10; Secondary: 14E07.

Received: April 24 2014;      Revised: January 23 2016;      Available Online: February 26 2016.