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

Abstract

We start the study of the family of birational maps $(f_{\alpha ,\beta })$ of ${\mathbb{P}}_{\mathbb{C}}^2$ 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 ${\mathbb{P}}_{\mathbb{C}}^1\times {\mathbb{P}}_{\mathbb{C}}^1$ 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 $f_{\alpha ,\beta }$.

Proof of Theorem A has been detailed

References in corpus (1)

• Complex one-frequency cocycles