The objective of this paper is to exhibit a relation of Fenchel-Nielsen coordinate system and the coordinate system by geodesic length functions for two-dimensional Teichmuller spaces. We express Fenchel-Nielsen coordinates by trace functions and prove Wolpert's formulas for hyperbolic sphere with four holes and torus with one hole. Furthermore, we consider the shortest of lengths of closed geodesics on these hyperbolic surfaces.
In the boundary of the Mandelbrot set, we can find quasiconformal copies of a Cantor Julia set which is a small perturbation of the Julia set of any given parameter in the boundary of the Mandelbrot set. We can also find copies that are images of the Julia sets by quasiconformal maps with dilatation arbitrarily close to 1. This answers a question by Adrian Douady. Indeed, we can specify the locations of such copies near the boundary of any small Mandelbrot set. If we zoom in the middle part of such a copy, then we can find a certain nested structure ("decoration") and finally another "smaller Mandelbrot set" appears. A similar nested structure exists in the Julia set for any parameter in the "smaller Mandelbrot set". All the parameters belonging to these quasiconformal copies in the Mandelbrot set are semihyperbolic and this leads to the fact that the set of semihyperbolic but non-Misiurewicz and non-hyperbolic parameters is dense with Hausdorff dimension 2 in the boundary of the Mandelbrot set.
In this paper we examine an orbit of simply connected wandering domains for the function f(z) = z cos z + 2 pi. They are noteworthy in that they are non-congruent but arise from a simple closed form function. Moreover, the shape of the wandering domains, suitably scaled, converges in the Hausdorff metric to the filled-in parabolic basin of the quadratic z(2) + c with c = 1/4, commonly named the "cauliflower". We complete our analysis by classifying the wandering domains within the Ann. 383 (2022), pp. 1127-1178], finding they are contracting and the diameters of the wandering domains tend to zero. To conclude we propose an expansion of the analysis to a wider family of functions and discuss some potential results.
In this paper, we extend the ideas of certain notions that one studies in thermodynamic formalism of maps to the context when the dynamics in the phase space evolves by complex holomorphic correspondences. Towards that end, we define the topological entropy of holomorphic correspondences using spanning sets. We then define the pressure of a real-valued continuous function defined on the Riemann sphere and investigate the Ruelle operator with respect to the Holder continuous function, however restricted on the support of the Dinh-Sibony measure.
Green's functions are highly useful in analyzing the dynamical behavior of polynomials in their escaping set. The aim of this paper is to construct an analogue of Green's functions for planar quasiregular mappings of degree two and constant complex dilatation. These Green's functions are dynamically natural, in that they semi-conjugate our quasiregular mappings to the real squaring map. However, they do not share the same regularity properties as Green's functions of polynomials. We use these Green's functions to investigate properties of the boundary of the escaping set and give several examples to illustrate behavior that does not occur for the dynamics of quadratic polynomials.
