Structural stability implies hyperbolicity and alpha-expansion for piecewise Möbius transformations

Let $F$ be a piecewise Möbius transformation (PMT), and let structural stability, hyperbolicity, and $b1$-expansion have their meanings defined in the paper. Structural-stability conjecture. If $F$ is a structurally stable PMT, then it is hyperbolic and $b1$-expanding. This is presented as an expected analogue of the corresponding result for rational maps; the paper gives no resolution of the conjecture.