  We give counterexamples to a question of Bowditch that if a non-elementary
type-preserving representation $\rho:\pi_1(\Sigma_{g,n})\rightarrow
PSL(2;\mathbb R)$ of a punctured surface group sends every non-peripheral
simple closed curve to a hyperbolic element, then must $\rho$ be Fuchsian. The
counterexamples come from relative Euler class $\pm1$ representations of the
four-punctured sphere group. We also show that the mapping class group action
on each non-extremal component of the character space of type-preserving
representations of the four-punctured sphere group is ergodic, which confirms a
conjecture of Goldman for this case. The main tool we use is Kashaev-Penner's
lengths coordinates of the decorated character spaces.
