Structural stability in piecewise M\"obius transformations

Abstract

Structural stability of piecewise M\"obius transformations (PMTs) is examined from various perspectives. A result concerning structural stability, restricted to the space of PMTs, is derived using hyperbolic characteristics of the component functions and the pre-singularities set, which facilitates a holomorphic motion. The analogous concept of J-stability for rational maps is defined and analyzed for PMTs, revealing some connections to general structural stability. The definitions of hyperbolic and expansive PMTs are introduced, demonstrating that they are not equivalent and that neither implies structural stability. By synthesizing the previous results and analyses, sufficient conditions for structural stability are established. Lastly, an example of structural stability within the tent maps family, extended to the complex plane, is presented.