We start the study of the family of birational maps $(f_{ \alpha,\beta})$ of $\mathbb{ P}^2_\mathbb{ C}$ in \cite{ Deserti}. For generic $\alpha$ and $\beta$ of modulus 1 the centraliser of $f_{ \alpha,\beta}$ is trivial, the topological entropy of $f_{ \alpha,\beta}$ is 0, there exist two areas 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 (\cite{ Avila}) we determine the \textsc{ Lyapunov} exponent of the cocyle associated to $f_{ \alpha,\beta}$.
Categories: math.DS