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.
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.
To investigate the degree d d connectedness locus, Thurston [On the geometry and dynamics of iterated rational maps, Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied σ d \sigma _d -invariant laminations, where σ d \sigma _d is the d d -tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f ( z ) = z 2 + c f(z) = z^2 +c . In the spirit of Thurston’s work, we consider the space of all cubic symmetric polynomials f λ ( z ) = z 3 + λ 2 z f_\lambda (z)=z^3+\lambda ^2 z in a series of three articles. In the present paper, the first in the series, we construct a lamination C s C L C_sCL together with the induced factor space S / C s C L \mathbb {S}/C_sCL of the unit circle S \mathbb {S} . As will be verified in the third paper of the series, S / C s C L \mathbb {S}/C_sCL is a monotone model of the cubic symmetric connectedness locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.
We consider moduli spaces of dynamical systems of correspondences over the projective line as a generalization of moduli spaces of dynamical systems of endomorphisms on the projective line. We define the moduli space D y n d , e Dyn_{d,e} of degree ( d , e ) (d,e) correspondences. We construct a family ρ c : D y n d , e ⇢ D y n 1 , d + e − 1 \rho _c : Dyn_{d,e} \dashrightarrow Dyn_{1,d+e-1} of rational maps representation-theoretically. Here we note that D y n 1 , d + e − 1 Dyn_{1,d+e-1} is identical to the moduli space of the usual dynamical systems of degree d + e − 1 d+e-1 . We show that the moduli space D y n d , e Dyn_{d,e} is rational by using ρ c \rho _c . Moreover, the multiplier maps for the fixed points factor through ρ c \rho _c . Furthermore, we show the Woods Hole formulae for correspondences of different degrees are also related by ρ c \rho _c and obtain another representation-theoretically simplified form of the formula.
According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group H H whose limit set is a generalized Apollonian gasket Λ H \Lambda _H . We design a surgery that relates H H to a rational map g g whose Julia set J g \mathcal {J}_g is (non-quasiconformally) homeomorphic to Λ H \Lambda _H . We show for a large class of triangulations, however, the groups of quasisymmetries of Λ H \Lambda _H and J g \mathcal {J}_g are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of H H , this group is equal to the group of Möbius symmetries of Λ H \Lambda _H , which is the semi-direct product of H H itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when Λ H \Lambda _H is the classical Apollonian gasket), we give a quasiregular model for the above actions which is quasiconformally equivalent to g g and produces H H by a David surgery. We also construct a mating between the group and the map coexisting in the same dynamical plane and show that it can be generated by Schwarz reflections in the deltoid and the inscribed circle.
Since the 1980s, much progress has been done in completely determining which functions share a Julia set. The polynomial case was completely solved in 1995, and it was shown that the symmetries of the Julia set play a central role in answering this question. The rational case remains open, but it was already shown to be much more complex than the polynomial one. Here, we offer partial extensions to Beardon’s results on the symmetry group of Julia sets, and discuss them in the context of singularly perturbed maps.
In this paper we study properties of endomorphisms of P k \mathbb {P}^k using a symmetric product construction ( P 1 ) k / S k ≅ P k (\mathbb {P}^1)^k/\mathfrak {S}_k \cong \mathbb {P}^k . Symmetric products have been used to produce examples of endomorphisms of P k \mathbb {P}^k with certain characteristics, k ≥ 2 k\geq 2 . In the present note, we discuss the use of these maps to enlighten stability phenomena in parameter spaces. In particular, we study k k -deep post-critically finite maps and characterize families of Lattès maps.
We study endomorphisms constructed by Sarah Koch in [Teichmüller theory and critically finite endomorphisms, Advances in Mathematics 248 (2013), 573–617] and we focus on the eigenvalues of the differential of such maps at its fixed points. To each post-critically finite unicritical polynomial, Koch associated a post-critically algebraic endomorphism of C P k {\mathbb {CP}}^k . Koch showed that the eigenvalues of the differentials of such maps along periodic cycles outside the post-critical sets have modulus strictly greater than 1 1 . In this article, we show that the eigenvalues of the differentials at fixed points are either 0 0 or have modulus strictly greater than 1 1 . This confirms a conjecture proposed by the author in his thesis. We also provide a concrete description of such values in terms of the multiplier of a unicritical polynomial.
