The group of automorphisms of the polynomial
is isomorphic to
For , the –action on displays rich and varied dynamics. The action of preserves a Poisson structure defining a –invariant area form on each . For , the action of is properly discontinuous on the four contractible components of and ergodic on the compact component (which is empty if ). The contractible components correspond to Teichmüller spaces of (possibly singular) hyperbolic structures on a torus . For , the level set consists of characters of reducible representations and comprises two ergodic components corresponding to actions of on and respectively. For , the action of on is ergodic. Corresponding to the Fricke space of a three-holed sphere is a –invariant open subset whose components are permuted freely by a subgroup of index in . The level set intersects if and only if , in which case the –action on the complement is ergodic.
"The modular group action on real $SL(2)$–characters of a one-holed torus." Geom. Topol. 7 (1) 443 - 486, 2003. https://doi.org/10.2140/gt.2003.7.443