Let \(M\) be a compact oriented surface of genus one with one boundary component. The polynomial \[ \kappa(x,y,z)=x^2+y^2+z^2-xyz-2 \] arises in the definition of the locus of the space of representations of the fundamental group of \(M\) in \(SU(2,\mathbb C)\). Let \(\Gamma\) be the polynomial automorphism of \(\mathbb C^3\) that preserves the polynomial \(\kappa\). The author gives a complete classification of the dynamics of the action of \(\Gamma\) on the fibres \(\kappa^{-1}(t)\cap \mathbb R^3\) for all \(t\) in \(\mathbb R\). This action of \(\Gamma\) preserves a Poisson structure defining a \(\Gamma-\)invariant area form on each \(\kappa^{-1}(t)\cap \mathbb R^3\). The main result of the paper is the following
Theorem :
\(\bullet\) For \(t<-2\), the group \(\Gamma\) acts properly on \(\kappa^{-1}(t)\cap \mathbb R^3\);
\(\bullet\) For \(-2\leq t< 2\), there is a compact connected component \(C_t\) of \(\kappa^{-1}(t)\cap \mathbb R^3\) and \(\Gamma\) and \(\Gamma\) acts properly on the complement \(\kappa^{-1}(t)\cap \mathbb R^3-C_t\);
\(\bullet\) For \(t=2\), the action of \(\Gamma\) is ergodic on the compact subset \(\kappa^{-1}(2)\cap[-2,3]^3\) and the action is ergodic on the complement \(\kappa^{-1}(2)- [-2,2]^3\);
\(\bullet\) For \(2<t\leq 18\), the group \(\Gamma\) acts ergodically on \(\kappa^{-1}(t)\cap \mathbb R^3\);
\(\bullet\) for \(t>18\), the group \(\Gamma\) acts properly and freely on an open subset \(\Omega_t\subset \kappa^{-1}(t)\cap \mathbb R^3\), permuting its components. The \(\Gamma-\) action on the complement of \(\Omega_t\) is ergodic.
By a classical result of Horowitz, the action of the group \(\Gamma\) is commensurable with the action of the modular group of \(M\) (which is seen as the group of automorphisms of \(\pi_1(M)\)) on the space of equivalence classes of representations \(\pi_1(M)\to SL(2,\mathbb C)\).
The author gives an identification of the properly discontinuous action of \(\Gamma\) on \(\kappa^{-1}(t)\cap \mathbb R^3\), for various real numbers \(t\), with actions of the modular group on various Teichmüller spaces of (possibly singular) hyperbolic structures on the surface \(M\). For \(t>2\), the discussion involves the Fricke space of the three-holed sphere.
The paper is very interesting and well-written.