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Recently, it is proven that positive harmonic functions defined in the unit disc or the upper half-plane in $\mathbb{C}$ are contractions in hyperbolic metrics [14]. Furthermore, the same result does not hold in higher dimensions as shown by given counterexamples [16]. In this paper, we shall show that positive (or bounded) harmonic functions defined in the unit ball in $\mathbb{R}^{n}$ are Lipschitz in hyperbolic metrics. The involved method in main results allows to establish essential improvements of Schwarz type inequalities for monogenic functions in Clifford analysis [24, 25] and octonionic analysis [21] in a unified approach.
We prove an effective estimate with a power saving error term for the number of square-tiled surfaces in a connected component of a stratum of quadratic differentials whose vertical and horizontal foliations belong to prescribed mapping class group orbits and which have at most L squares. This result strengthens asymptotic counting formulas in the work of Delecroix, Goujard, Zograf, Zorich, and the author.
The local invariants of a meromorphic quadratic differential on a compact Riemann surface are the orders of zeros and poles, and the residues at the poles of even orders. The main result of this paper is that with few exceptions, every pattern of local invariants can be obtained by a quadratic differential on some Riemann surface. The exceptions are completely classified and only occur in genera zero and one. Moreover, in the case of a nonconnected stratum, we show that, with three exceptions in genus one, each configuration of invariants can be realized in each non-hyperelliptic connected component of the stratum. In the hyperelliptic components with two poles the residues at both poles coincide. These results are obtained using the flat metric induced by the differentials. We give an application by bounding the number of disjoint cylinders on a primitive quadratic differential.
Let ${{\mathcal {H}}}$ be a stratum of translation surfaces with at least two singularities, let $m_{{{\mathcal {H}}}}$ denote the Masur-Veech measure on ${{\mathcal {H}}}$, and let $Z_0$ be a flow on $({{\mathcal {H}}}, m_{{{\mathcal {H}}}})$ obtained by integrating a Rel vector field. We prove that $Z_0$ is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces $({\mathcal L}, m_{{\mathcal L}})$, where ${\mathcal L} \subset {{\mathcal {H}}}$ is an orbit-closure for the action of $G = \operatorname {SL}_2({\mathbb {R}})$ (i.e., an affine invariant subvariety) and $m_{{\mathcal L}}$ is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of $Z_0$ with respect to any of the measures $m_{{{\mathcal L}}}$ is zero.
The central theme of this paper is the holomorphic spectral theory of the canonical Laplace operator of the complement of the “complexified unit circle” $\{(z,w) \in \widehat {{\mathbb C}}^2 \colon z \cdot w = 1\}$. We start by singling out a distinguished set of holomorphic eigenfunctions on the bidisk in terms of hypergeometric ${}_2F_1$ functions and prove that they provide a spectral decomposition of every holomorphic eigenfunction on the bidisk. As a second step, we identify the maximal domains of definition of these eigenfunctions and show that these maximal domains naturally determine the fine structure of the eigenspaces. Our main result gives an intrinsic classification of all closed Möbius invariant subspaces of eigenspaces of the canonical Laplacian of $\Omega $. Generalizing foundational prior work of Helgason and Rudin, this provides a unifying complex analytic framework for the real-analytic eigenvalue theories of both the hyperbolic and spherical Laplace operators on the open unit disk resp. the Riemann sphere and, in particular, shows how they are interrelated with one another.
We investigate the translation lengths of group elements that arise in random walks on the isometry groups of Gromov hyperbolic spaces. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov, and almost every random walk on $\mathrm {Out}(F_n)$ eventually becomes fully irreducible. If the underlying measure further has finite first moment, then the growth rate of translation lengths is equal to the drift, the escape rate of the random walk.
We then apply our technique to investigate the random walks induced by the action of mapping class groups on Teichmüller spaces. In particular, we prove the spectral theorem under finite first moment condition, generalizing a result of Dahmani and Horbez.
We show convergence of small eigenvalues for geometrically finite hyperbolic n-manifolds under strong limits. For a class of convergent convex sets in a strongly convergent sequence of Kleinian groups, we use the spectral gap of the limit manifold and the exponentially mixing property of the geodesic flow along the strongly convergent sequence to find asymptotically uniform counting formulas for the number of orthogeodesics between the convex sets. In particular, this provides asymptotically uniform counting formulas (with respect to length) for orthogeodesics between converging Margulis tubes, geodesic loops based at converging basepoints, and primitive closed geodesics.
Assume that f is a real ρ-harmonic function of the unit disk $\mathbb{D}$ onto the interval $(-1,1)$, where $\rho(u,v)=R(u)$ is a metric defined in the infinite strip $(-1,1)\times \mathbb{R}$. Then we prove that $|\nabla f(z)|(1-|z|^2)\le \frac{4}{\pi}(1-f(z)^2)$ for all $z\in\mathbb{D}$, provided that ρ has a non-negative Gaussian curvature. This extends several results in the field and answers to a conjecture proposed by the first author in 2014. Such an inequality is not true for negatively curved metrics.
Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including Liouville property) between Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai's hypotheses in many cases are not satisfied in the context of Riemann surfaces endowed with the Poincaré metric. In this work we fill that gap for the Liouville property, by proving its stability by quasi-isometries for every Riemann surface (and even Riemannian surfaces with pinched negative curvature). Also, a key result characterizes Riemannian surfaces which are quasi-isometric to $\mathbb {R}$.
This paper is about a type of quantitative density of closed geodesics and orthogeodesics on complete finite-area hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic and the shortest doubly truncated orthogeodesic that are $\varepsilon$-dense on a given compact set on the surface.
We prove that the hitting measure is singular with respect to the Lebesgue measure for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, the Hausdorff dimension of the hitting measure is strictly less than one. Equivalently, the inequality between entropy and drift is strict. A similar statement is proven for Coxeter groups.
Recall that two geodesics in a negatively curved surface S are of the same type if their free homotopy classes differ by a homeomorphism of the surface. In this note we study the distribution in the unit tangent bundle of the geodesics of fixed type, proving that they are asymptotically equidistributed with respect to a certain measure
${\mathfrak {m}}^S$
on
$T^1S$
. We study a few properties of this measure, showing for example that it distinguishes between hyperbolic surfaces.
In this article, we establish an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps. The correspondence, which is expressed using simple planar graphs, has several dynamical consequences. As an application of this correspondence, we give complete answers to geometric mating problems for critically fixed anti-rational maps.
Given an integer
$g>2$
, we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.
Consider a component
${\cal Q}$
of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property
${\cal P}$
for periodic orbits of the Teichmüller flow on
${\cal Q}$
typical if the growth rate of orbits with property
${\cal P}$
is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle
$V\to {\cal Q}$
, the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.
This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work by Goldberg and Keen [The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift. Invent. Math.101(2) (1990), 335–372] on degree two rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further is that, despite the presence of the essential singularity, our new functions exhibit a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted.
In 1955, Lehto showed that, for every measurable function
$\psi $
on the unit circle
$\mathbb T,$
there is a function f holomorphic in the unit disc, having
$\psi $
as radial limit a.e. on
$\mathbb T.$
We consider an analogous problem for solutions f of homogenous elliptic equations
$Pf=0$
and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.
In this paper we investigate the range of validity of Ruelle’s property. First, we show that every finitely generated Fuchsian group has Ruelle’s property. We also prove the existence of an infinitely generated Fuchsian group satisfying Ruelle’s property. Concerning the negative results, we first generalize Astala and Zinsmeister’s results [Mostow rigidity and Fuchsian groups. C. R. Math. Acad. Sci. Paris311 (1990), 301–306; Teichmüller spaces and BMOA. Math. Ann.289 (1991), 613–625] by proving that all convergence-type Fuchsian groups of the first kind fail to have Ruelle’s property. Finally, we give some results about second-kind Fuchsian groups. [-3.2pc]
We associate a flow
$\phi $
with a solution of the vortex equations on a closed oriented Riemannian 2-manifold
$(M,g)$
of negative Euler characteristic and investigate its properties. We show that
$\phi $
always admits a dominated splitting and identify special cases in which
$\phi $
is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of
$(M,g)$
.