Automorphisms of Two-Generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane

The automorphisms of a two-generator free group F-2 acting on the space of orientation-preserving isometric actions of F-2 on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action of a group Gamma on R-3 by polynomial automorphisms preserving the cubic polynomial kappa(Phi)(x, y, z) := -x(2) - y(2) + z(2) + xyz - 2 and an area form on the level surfaces kappa(-1)(Phi) (k). The Fricke space of marked hyperbolic structures on the 2-holed projective plane with funnels or cusps identifies with the subset F(C-0,C-2) subset of R-3 defined by z <= -2, xy+z >= 2. The generalized Fricke space of marked hyperbolic structures on the 1-holed Klein bottle with a funnel, a cusp, or a conical singularity identifies with the subset F' (C-1,C-1) subset of R-3 defined by z > 2, xyz >= x(2) + y(2). We show that Gamma acts properly on the subsets Gamma.F(C-0,C-2) and Gamma.F' (C-1,C-1). Furthermore for each k < 2, the action of Gamma is ergodic on the complement of Gamma.F(C-0,C-2) in kappa(-1)(Phi) (k) for k < -14. In particular, the action is ergodic on all of kappa(-1)(Phi) (k) for -14 <= k < 2. For k > 2, the orbit Gamma.F(C-1,C-1) is open and dense in kappa(-1)(Phi)(k). We conjecture its complement has measure zero.