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.
We prove a suite of dynamical results, including exactness of the transformation and piecewise-analyticity of the invariant measure, for a family of continued fraction systems, including specific examples over reals, complex numbers, quaternions, octonions, and in $R^3$. Our methods expand on the work of Nakada and Hensley, and in particular fill some gaps in Hensley's analysis of Hurwitz complex continued fractions. We further introduce a new ``serendipity'' condition for a continued fraction algorithm, which controls the long-term behavior of the boundary of the fundamental domain under iteration of the continued fraction map, and which is under reasonable conditions equivalent to the finite range property. We also show that the finite range condition is extremely delicate: perturbations of serendipitous systems by non-quadratic irrationals do not remain serendipitous, and experimental evidence suggests that serendipity may fail even for some rational perturbations.
We consider circles on a translation surface X, consisting of points joined to a common center point by a geodesic of length R. We show that as R →∞ these circles distribute to a measure on X which is equivalent to the area. In the last section we consider analogous results for closed geodesics.
The invariant class under parabolic and near-parabolic renormalizations constructed by Inou and Shishikura has been proved to be extremely useful in recent years. It leads to several important progresses on the dynamics of certain holomorphic maps with critical points of local degree two. In this paper, we construct a new class consisting of holomorphic maps with critical points of local degree three which is invariant under parabolic and near-parabolic renormalizations. As potential applications, some results of cubic unicritical polynomials can be obtained similarly as the quadratic case. For example, the existence of cubic unicritical Julia sets with positive area, the characterizations of the topology and geometry of cubic irrationally indifferent attractors etc.
We investigate the distortion of the Assouad dimension and (regularized) spectrum of sets under planar quasiregular maps. The respective results for the Hausdorff and upper box-counting dimension follow immediately from their quasiconformal counterparts by employing elementary properties of these dimension notions (e.g. countable stability and Lipschitz stability). However, the Assouad dimension and spectrum do not share such properties. We obtain upper bounds on the Assouad dimension and spectrum of images of compact sets under holomorphic and planar quasiregular maps by studying their behavior around their critical points. As an application, the invariance of porosity of compact subsets of the plane under quasiregular maps is established.
The Teichmuller space of closed Riemann surfaces of genus two is considered as a subset of R 7 . We present some mapping classes acting on the model of the Teichuller space and investigate its action especially on the extremal surfaces with respect to a certain metric. As a result we estimate the distance between two orbits of distinct extremal surfaces from above.
We consider the family f(a,b)(x, y) = (y, (y + a)/(x + b)) of birational maps of the plane and the parameter values (a, b) for which fa, b gives an automorphism of a rational surface. In particular, we find values for which fa, b is an automorphism of positive entropy but no invariant curve. The Main Theorem: If fa, b is an automorphism with an invariant curve and positive entropy, then either (1) (a, b) is real, and the restriction of f to the real points has maximal entropy, or (2) fa, b has a rotation (Siegel) domain.
In the space $\mathcal{H}^2$ of hyperbolic surfaces decorated with a base unit vector, the topology induced by the Gromov-Hausdorff convergence coincides with the Chabauty topology on the space of discrete torsion-free subgroups of $\rm{PSL}_2(\mathbb{R})$. Using paths constructed from changing the Fenchel-Nielsen coordinates and shrinking simple closed curves to cusps, we demonstrate path-connectivity of $\mathcal{H}^2$ and some of its subspaces.
In this paper, we analyze a certain family of holomorphic correspondences on C ^ × C ^ \hat {\mathbb {C}}\times \hat {\mathbb {C}} and prove their equidistribution properties. In particular, for any correspondence in this family we prove that the naturally associated multivalued map F F is such that for any a ∈ C a\in \mathbb {C} , we have that ( F n ) ∗ ( δ a ) (F^n)_*(\delta _a) converges to a probability measure μ F \mu _F for which F ∗ ( μ F ) = μ F d F_*(\mu _F)=\mu _F d where d d is the degree of F F . This result is used to show that the minimal Hutchinson invariant set, introduced by P. Alexandersson, P. Brändén, and B. Shapiro [An inverse problem in Pólya–Schur theory. I. Non-degenerate and degenerate operators, preprint, 2024], of a large class of operators and for sufficiently large n n exists and is the support of the aforementioned measure. We prove that under a minor additional assumption, the minimal Hutchinson-invariant set is a Cantor set.
In this work we show that every quotient of a torus endomorphism has a parabolic orbifold, answering a question of Mario Bonk and Daniel Meyer posed in [ Expanding Thurston maps , American Mathematical Society, Providence, RI, 2017].
In this article, for degree d >= 1, we construct an embedding Phi d of the connectedness locus M d +1 of the polynomials z d + 1 + c into the connectedness locus of degree 2d + 1 bicritical odd polynomials.
This paper is part of a general program in complex dynamics to understand parameter spaces of transcendental maps with finitely many singular values. The simplest families of such functions have two asymptotic values and no critical values. These families, up to affine conjugation, depend on two complex parameters. Understanding their parameter spaces is key to understanding families with more asymptotic values, just as understanding quadratic polynomials was for rational maps more generally. The first such families studied were the one-dimensional slices of the exponential family, exp ( z ) + a \exp (z) + a , and the tangent family λ tan z \lambda \tan z . The exponential case exhibited phenomena not seen for rational maps: Cantor bouquets in both the dynamic and parameter spaces, and no bounded hyperbolic components. The tangent case, with its two finite asymptotic values ± λ i \pm \lambda i , is closer to the rational case, a kind of infinite degree version of the latter. In this paper, we consider a general family that interpolates between exp ( z ) + a \exp (z) + a and λ tan z \lambda \tan z . Our new family has two asymptotic values and a one-dimensional slice for which one of the asymptotic values is constrained to be a pole, the “polar asymptotic value” of the title. We show how the dynamic and parameter planes for this slice exhibit behavior that is a surprisingly delicate interplay between that of the exp ( z ) + a \exp (z) + a and λ tan z \lambda \tan z families.
We analyze the existence and types of unbounded Fatou components for elliptic functions and other meromorphic functions with doubly periodic Julia sets. We show that apart from Herman rings and Siegel disks, all types of dynamics can occur in these domains, which are called toral bands. We show that toral bands are not necessarily periodic, and we give results about the number of distinct residue classes of critical points in each toral band.
In the paper [The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479], Wolf provided a global coordinate system of the Teichmüller space of a closed oriented surface S S with the vector space of holomorphic quadratic differentials on a Riemann surface X X homeomorphic to S S . This coordinate system is via harmonic maps from the Riemann surface X X to hyperbolic surfaces. Moreover, he gave a compactification of the Teichmüller space by adding a point at infinity to each endpoint of harmonic map rays starting from X X in the space. Wolf also showed this compactification coincides with the Thurston compactification. In this paper, we extend the harmonic map ray compactification to the case of punctured Riemann surfaces and show that it still coincides with the Thurston compactification.
We discuss problems concerning the conformal condenser capacity of “hedgehogs”, which are compact sets E E in the unit disk D = { z : | z | > 1 } \mathbb {D}=\{z:\,|z|>1\} consisting of a central body E 0 E_0 that is typically a smaller disk D ¯ r = { z : | z | ≤ r } \overline {\mathbb {D}}_r=\{z:\,|z|\le r\} , 0 > r > 1 0>r>1 , and several spikes E k E_k that are compact sets lying on radial intervals I ( α k ) = { t e i α k : 0 ≤ t > 1 } I(\alpha _k)=\{te^{i\alpha _k}:\,0\le t>1\} . The main questions we are concerned with are the following: (1) How does the conformal capacity c a p ( E ) \mathrm {cap}(E) of E = ∪ k = 0 n E k E=\cup _{k=0}^n E_k behave when the spikes E k E_k , k = 1 k=1 , …, n n , move along the intervals I ( α k ) I(\alpha _k) toward the central body if their hyperbolic lengths are preserved during the motion? (2) How does the capacity c a p ( E ) \mathrm {cap}(E) depend on the distribution of angles between the spikes E k E_k ? We prove several results related to these questions and discuss methods of applying symmetrization type transformations to study the capacity of hedgehogs. Several open problems, including problems on the capacity of hedgehogs in the three-dimensional hyperbolic space, will also be suggested.
This paper, which is the second of a series of three papers, studies dynamical properties of elements of Out ( F n ) \operatorname {Out}(F_{\mathtt n}) , the outer automorphism group of a nonabelian free group F n F_{\mathtt n} . We prove that, for every exponentially growing outer automorphism of F n F_{\mathtt n} , there exists a preferred compact topological space, the space of currents relative to a malnormal subgroup system, on which ϕ \phi acts by homeomorphism with a North-South dynamics behavior.
