The metric geometry of generalized pseudo-Anosov maps - Vasconcelos, Luciana Menezes (2024)
The article [dC05] introduced the notion of generalized pseudo-Anosov (gpA) maps, which extend the pseudo-Anosov (pA) transformations introduced by Thurston by allowing the presence of an infinite number of singularities, with only a finite number of accumulation points. Similar to the pseudo-Anosov case, the conic-flat structure persists away from the accumulation points, and it is possible to prove [dCH12] that the induced complex structure extends uniquely over these points. In [BdCH21], a continuous family of sphere homeomorphisms was constructed as a quotient of the inverse limit of the tent family of interval endomorphisms. This family includes the unimodal generalized pseudo-Anosov (ugpA) maps as a countable, dense subfamily, whose spheres of definition have well-defined and extensively studied geometric structures. In this work, we examine the family of unimodal generalized pseudo-Anosov maps, exploring their unique geometric and dynamical properties. We show that the associated surfaces exhibit several remarkable characteristics, such as Ahlfors regularity and linear local contractibility, which are crucial in understanding the larger context of quasisymmetric structures on metric surfaces. Consequently, by applying a theorem by Bonk and Kleiner [BK01], we establish that these surfaces are quasisymmetrically equivalent to the topological 2-sphere.