Structural stability in piecewise Möbius transformations

TL;DR

The paper studies structural stability for piecewise Möbius transformations (PMTs) on the Riemann sphere, introducing hyperbolicity, $\alpha$-expansivity, and a rational-map analogue of J-stability. It proves sufficient conditions for PMT structural stability inside $PSL(2,\mathbb{C})^K$ via a holomorphic motion of the relevant invariant sets and Bers-Royden extension, without relying on the stability of the generated Möbius group. A $\mathcal{B}$-stability notion is developed to track holomorphic deformations of the discontinuity set, revealing that $\mathcal{B}$-stability implies stability on the $\alpha$-limit set but need not yield full topological conjugacy. The work also demonstrates stability in concrete families, notably complex tent maps, under explicit parameter constraints that guarantee hyperbolicity and $\alpha$-expansion. Overall, the results bridge holomorphic deformation techniques with PMT dynamics, providing concrete criteria for when PMTs maintain qualitative behavior under perturbations and offering a framework for further exploration of stability in piecewise complex dynamics.

Abstract

Structural stability of piecewise Möbius 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.

Figures (5)

• Figure 1: The pre-discontinuity and regular sets of $F_{\lambda}$ described in Example \ref{['example:HiperNoSS']}. Left: With $\lambda=\frac{1}{2}-0.223i$. $R_{1}$ is the immediate basin of attraction of $z_{\lambda}$. Right: With $\lambda=\frac{1}{2}-(0.223+\varepsilon)i$, $0<\varepsilon\ll1$. $R_{1}$ contains several regular components.
• Figure 2: Pre-discontinuity set (depicted in black) and regular set (drawn with colors) of $F$ from example \ref{['example:ExpandNoHyper']}.
• Figure 3: The pre-discontinuity and regular sets of $F$ composed of $f_{1}$ and $f_{2}$, from Example \ref{['example:SS']}. Top left: $f_{1}(z)=\frac{(1+i)z+1.02i}{-1.02iz+(1-i)}$ and $f_{2}(z)=\frac{(1+i)z-1.02i}{1.02iz+(1-i)}$. Top right: $f_{1}(z)=\frac{(1+i)z+c}{-cz+(1-i)}$ and $f_{2}(z)=\frac{(1+i)z-c}{cz+(1-i)}$, with $c=0.99+0.01i$. Bottom left: $f_{1}(z)=\frac{(1+i)z+i}{-iz+(1-i)}$ and $f_{2}(z)=\frac{(0.8+i)z-i}{iz+(1-i)}$. Bottom right: $f_{1}(z)=\frac{(1+i)z+i}{-iz+(1-i)}$ and $f_{2}(z)=\frac{(1.1+i)z+0.1-i}{(-0.1+i)z+(0.9-i)}$.
• Figure 4: Holomorphic motion of $\mathcal{B}(F_{0,i})$, from Example \ref{['example:HoloMot']}. Left: With $\mu=0$ and $\lambda=i$, $F_{0,i}$ has a unique fixed point at $z=1$, which is parabolic. Right: With $\mu\approx0$ and $\lambda\approx i$, $F_{\mu,\lambda}$ has two fixed points: $\frac{i+\sqrt{-1-\lambda{{}^2}}}{\lambda}$ attracting, and $\frac{i-\sqrt{-1-\lambda{{}^2}}}{\lambda}$ repelling.
• Figure 5: Pre-discontinuity and regular sets of the tent maps $T_{B,\lambda}$ from Example \ref{['example:TentSS']}. Top left: With $\lambda=\frac{7}{2}$. Top right: With $\lambda=2+2i$. Bottom left: With $\lambda=\frac{11}{4}i$. Bottom right: With $\lambda=-\frac{5}{2}+2i$.

Theorems & Definitions (68)

• Definition 1
• Remark 1
• Definition 2
• Definition 3
• Remark 2
• Definition 4
• Remark 3
• Remark 4
• Definition 5
• Definition 6