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The boundary $\partial X$ of a boundary continuous Gromov hyperbolic space $X$ carries a natural Moebius structure on the boundary. For a proper, geodesically complete, boundary continuous Gromov hyperbolic space $X$, the boundary $\partial X$ equipped with its cross-ratio is a particular kind of quasi-metric space, called a quasi-metric antipodal space. Given a quasi-metric antipodal space $Z$, one may consider the family of all hyperbolic fillings of $Z$. In \cite{biswas2024quasi} it was shown that this family has a unique upper bound $\mathcal{M}(Z)$ (with respect to a natural partial order on hyperbolic fillings of $Z$), which can be described explicitly in terms of the cross-ratio on $Z$. As shown in \cite{biswas2024quasi}, the spaces $\mathcal{M}(Z)$ constitute a natural class of spaces called maximal Gromov hyperbolic spaces. A natural problem is to describe explicitly the maximal Gromov hyperbolic spaces $X$ whose boundary $\partial X$ is finite. We show that for a maximal Gromov hyperbolic space $X$ with boundary $\partial X$ of cardinality $n$, the space $X$ is isometric to a finite polyhedral complex embedded in $(\mathbb{R}^n, ||\cdot||_{\infty})$ with cells of dimension at most $n/2$, given by attaching $n$ half-lines to vertices of a compact polyhedral complex. In particular the geometry at infinity of $X$ is trivial. The combinatorics of the polyhedral complex is determined by certain relations $R \subset \partial X \times \partial X$ on the boundary $\partial X$, called antipodal relations. In \cite{biswas2024quasi} it was shown that maximal Gromov hyperbolic spaces are injective metric spaces. We give a shorter, simpler proof of this fact in the case of spaces with finite boundary. We also consider the space of deformations of a maximal Gromov hyperbolic space with finite boundary, and define an associated Teichmuller space.

Let $M$ be an $n$-dimensional Hadamard manifold of pinched negative curvature $-b^2 \leq K_M \leq -a^2$. The solution of the Dirichlet problem at infinity for $M$ leads to the construction of a family of mutually absolutely continuous probability measures $\{\mu_x\}_{x \in M}$ called the harmonic measures. Fixing a basepoint $o \in M$, the Poisson kernel of $M$ is the function $P : M \times \partial M \to (0, \infty)$ defined by \begin{equation*} P(x, \xi) = \frac{d\mu_x}{d\mu_o}(\xi) \ , \ x \in M, \xi \in \partial M. \end{equation*} We prove the following global upper and lower bounds for the Poisson kernel: \begin{equation*} \frac{1}{C}\: e^{-2K{(o|\xi)}_x}\: e^{a d(x, o)} \le P(x,\xi) \le C\: e^{2K{(x|\xi)}_o}\: e^{-a d(x,o)} \:, \end{equation*} for some positive constants $C \geq 1, K > 0$ depending solely on $a, b$ and $n$. The above estimates may be viewed as a generalization of the well-known formula for the Poisson kernel in terms of Busemann functions for the special case of Gromov hyperbolic harmonic manifolds. These estimates do not follow directly from known estimates on Green's functions or harmonic measures. Instead we use techniques due to Anderson-Schoen for estimating positive harmonic functions in cones. As applications, we obtain quantitative estimates for the convergence $\mu_x \to \delta_{\xi}$ as $x \in M \to \xi \in \partial M$, and for the convergence of harmonic measures on finite spheres to the harmonic measures on the boundary at infinity as the radius of the spheres tends to infinity., Comment: 22 pages, 3 figures

A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP). While this has so far been studied mainly for domains in $\mathbb{R}^n$, we consider this problem in the general setting of domains in Riemannian manifolds, and obtain results generalizing classical results of Fenton. We also obtain a result for complete, simply connected Riemannian manifolds of pinched negative curvature where there is no restriction on the radius function in the RMVP.

Hyperbolic fillings of metric spaces are a well-known tool for proving results on extending quasi-Moebius maps between boundaries of Gromov hyperbolic spaces to quasi-isometries between the spaces. For CAT(-1) spaces, and more generally boundary continuous Gromov hyperbolic spaces, one can refine the quasi-Moebius structure on the boundary to a Moebius structure. It is then natural to ask whether there exists a functorial hyperbolic filling of the boundary by a boundary continuous Gromov hyperbolic space with an identification between boundaries which is not just quasi-Moebius, but in fact Moebius. We give a positive answer to this question for a large class of boundaries satisfying one crucial hypothesis, the {\it antipodal property}. This gives a class of compact spaces called {\it quasi-metric antipodal spaces}. For any such space $Z$, we give a functorial construction of a boundary continuous Gromov hyperbolic space $\mathcal{M}(Z)$ together with a Moebius identification of its boundary with $Z$. The space $\mathcal{M}(Z)$ is maximal amongst all fillings of $Z$. These spaces $\mathcal{M}(Z)$ give in fact all examples of a natural class of spaces called {\it maximal Gromov hyperbolic spaces}. We prove an equivalence of categories between quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces. This is part of a more general equivalence we prove between the larger categories of certain spaces called {\it antipodal spaces} and {\it maximal Gromov product spaces}. We prove that the injective hull of a Gromov product space $X$ is isometric to the maximal Gromov product space $\mathcal{M}(Z)$, where $Z$ is the boundary of $X$. We also show that a Gromov product space is injective if and only if it is maximal., Comment: Introduction rewritten, added results on Gromov product spaces and injective metric spaces

The purpose of this note is to give details for an argument of Sullivan to construct eigenfunctions of the Laplacian on a Riemannian manifold using exit times of Brownian motion \cite{sullivanpos}. Let $X$ be a complete, simply connected Riemannian manifold of pinched negative sectional curvature. Let $\lambda_1 = \lambda_1(X) < 0$ be the supremum of the spectrum of the Laplacian on $L^2(X)$, and let $D \subset X$ be a bounded domain in $X$ with smooth boundary. Let $(B_t)_{t \geq 0}$ be Brownian motion on $X$ and let $\tau = \tau_D$ be the first exit time of Brownian motion from $D$. For each $\lambda \in \mathbb{C}$ with $\hbox{Re } \ \lambda > \lambda_1$ and $x \in D$, we show that for any continuous function $\phi : \partial D \to \mathbb{C}$, the function $$ h(x) = \mathbb{E}_x(e^{-\lambda \tau} \phi(B_{\tau})) \ , \ x \in D, $$ is an eigenfunction of the Laplacian on $D$ with eigenvalue $\lambda$ and boundary value $\phi$.

Let $X$ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces. Denote by $h > 0$ the mean curvature of horospheres in $X$, and set $\rho = h/2$. Fixing a basepoint $o \in X$, for $\xi \in \partial X$, denote by $B_{\xi}$ the Busemann function at $\xi$ such that $B_{\xi}(o) = 0$. then for $\lambda \in \C$ the function $e^{(i\lambda - \rho)B_{\xi}}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(\lambda^2 + \rho^2)$. For a function $f$ on $X$, we define the Fourier transform of $f$ by $$\tilde{f}(\lambda, \xi) := \int_X f(x) e^{(-i\lambda - \rho)B_{\xi}(x)} dvol(x)$$ for all $\lambda \in \C, \xi \in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X} \tilde{f}(\lambda, \xi) e^{(i\lambda - \rho)B_{\xi}(x)} d\lambda_o(\xi) |c(\lambda)|^{-2} d\lambda$$ for $f \in C^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$, $\lambda_o$ is the visibility measure on $\partial X$ with respect to the basepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon., Comment: arXiv admin note: substantial text overlap with arXiv:1802.07236

We give an introduction to the transalgebraic theory of simply connected log-Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves of genus $0$). We define the base vector space of transcendental functions and establish by elementary means some transcendental properties. We introduce the Ramificant Determinant constructed with transcendental periods and we give a closed-form formula that gives the main applications to transalgebraic curves. We prove an Abel-like Theorem and a Torelli-like Theorem. Transposing to the transalgebraic curve the base vector space of transcendental functions, they generate the structural ring from which the points of the transalgebraic curve can be recovered algebraically, including infinite ramification points., Comment: see also arXiv:1512.03776

Let $X, Y$ be complete, simply connected Riemannian surfaces with pinched negative curvature $-b^2 \leq K \leq -1$. We show that if $f : \partial X \to \partial Y$ is a Moebius homeomorphism between the boundaries at infinity of $X, Y$, then $f$ extends to an isometry $F : X \to Y$. This can be viewed as a generalization of Otal's marked length spectrum rigidity theorem for closed, negatively curved surfaces, in the sense that Otal's theorem asserts that if $X, Y$ admit properly discontinuous, cocompact, free actions by groups of isometries and the boundary map $f$ is Moebius and equivariant with respect to these actions then it extends to an isometry. In our case there are no cocompactness or equivariance assumptions, indeed the isometry groups of $X, Y$ may be trivial.

Let $(X, g_0)$ be a complete, simply connected Riemannian manifold with sectional curvatures $K_{g_0}$ satisfying $-b^2 \leq K_{g_0} \leq -1$ for some $b \geq 1$. Let $g_1$ be a Riemannian metric on $X$ such that $g_1 = g_0$ outside a compact in $X$, and with sectional curvatures $K_{g_1}$ satisfying $K_{g_1} \leq -1$. The identity map $id : (X, g_0) \to (X, g_1)$ is bi-Lipschitz, and hence induces a homeomorphism between the boundaries at infinity of $(X, g_0)$ and $(X, g_1)$, which we denote by $\hat{id}_{g_0, g_1} : \partial_{g_0} X \to \partial_{g_1} X$. We show that if the boundary map $\hat{id}_{g_0, g_1}$ is Moebius (i.e. preserves cross-ratios), then it extends to an isometry $F : (X, g_0) \to (X, g_1)$.

Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K \leq -1$. In \cite{biswas6}, a Fourier transform was defined for functions on $X$, and a Fourier inversion formula and Plancherel theorem were proved. We use the Fourier transform to investigate the dynamics on $L^p(X)$ for $p > 2$ of certain bounded linear operators $T : L^p(X) \to L^p(X)$ which we call "$L^p$-multipliers" in accordance with standard terminology. These operators are required to preserve the subspace of $L^p$ radial functions. A notion of convolution with radial functions was defined in \cite{biswas6}, and these operators are also required to be compatible with convolution in the sense that $$ T\phi * \psi = \phi * T\psi $$ for all radial $C^{\infty}_c$-functions $\phi, \psi$. They are also required to be compatible with translation of radial functions. Examples of $L^p$-multipliers are given by the operator of convolution with an $L^1$ radial function, or more generally convolution with a finite radial measure. In particular elements of the heat semigroup $e^{t\Delta}$ act as multipliers. Given $2 < p < \infty$, we show that for any $L^p$-multiplier $T$ which is not a scalar multiple of the identity, there is an open set of values of $\nu \in \mathbb{C}$ for which the operator $\frac{1}{\nu} T$ is chaotic on $L^p(X)$ in the sense of Devaney, i.e. topologically transitive and with periodic points dense. Moreover such operators are topologically mixing. We also show that there is a constant $c_p > 0$ such that for any $c \in \mathbb{C}$ with $\Re c > c_p$, the action of the shifted heat semigroup $e^{ct} e^{t\Delta}$ on $L^p(X)$ is chaotic. These results generalize the corresponding results for rank one symmetric spaces of noncompact type and negatively curved harmonic $NA$ groups (or Damek-Ricci spaces).

Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K \leq -1$, and let $\partial X$ denote the boundary at infinity of $X$. Let $h > 0$ denote the mean curvature of horospheres in $X$, and let $\rho = h/2$. Fixing a basepoint $o \in X$, for $\xi \in \partial X$, let $B_{\xi}$ denote the Busemann function at $\xi$ such that $B_{\xi}(o) = 0$, then for $\lambda \in \mathbb{C}$ the function $e^{(i\lambda - \rho)B_{\xi}}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(\lambda^2 + \rho^2)$. For a function $f$ on $X$, we define the Fourier transform of $f$ by $$\tilde{f}(\lambda, \xi) := \int_X f(x) e^{(-i\lambda - \rho)B_{\xi}(x)} dvol(x)$$ for all $\lambda \in \mathbb{C}, \xi \in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X} \tilde{f}(\lambda, \xi) e^{(i\lambda - \rho)B_{\xi}(x)} d\lambda_o(\xi) |c(\lambda)|^{-2} d\lambda$$ for $f \in C^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$, $\lambda_o$ is the visibility measure on $\partial X$ with respect to the basepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Plancherel theorem. This generalizes the corresponding results for rank one symmetric spaces of noncompact type and negatively curved harmonic $NA$ groups (or Damek-Ricci spaces)., Comment: 32 pages

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, and a family of probability measures $\{ \mu_x \}_{x \in X}$ on $\partial X$, we describe a continuous family of extensions $\{\hat{f}_p : X \to Y \}_{1 \leq p \leq \infty}$ of $f$, called the hyperbolic $p$-barycenter maps of $f$. If all the measures $\mu_x$ have full support then for $p = \infty$ the map $\hat{f}_{\infty}$ coincides with the circumcenter map $\hat{f}$ defined previously in \cite{biswas5}. We use this to show that if $X, Y$ are complete, simply connected manifolds with sectional curvatures $K$ satisfying $-b^2 \leq K \leq -1$, then the circumcenter maps of $f$ and $f^{-1}$ are $\sqrt{b}$-bi-Lipschitz homeomorphisms which are inverses of each other. It follows that closed negatively curved manifolds with the same marked length spectrum are bi-Lipschitz homeomorphic., Comment: 31 pages. arXiv admin note: text overlap with arXiv:1709.09110

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, we describe an extension $\hat{f} : X \to Y$ of $f$, called the circumcenter map of $f$, which is constructed using circumcenters of expanding sets. The extension $\hat{f}$ is shown to coincide with the $(1, \log 2)$-quasi-isometric extension constructed in [biswas3], and is locally $1/2$-Holder continuous. When $X,Y$ are complete, simply connected manifolds with sectional curvatures $K$ satisfying $-b^2 \leq K \leq -1$ for some $b \geq 1$ then the extension $\hat{f} : X \to Y$ is a $(1, (1 - \frac{1}{b})\log 2)$-quasi-isometry. Circumcenter extension of Moebius maps is natural with respect to composition with isometries., Comment: Added local Holder continuity of circumcenter extension. Minor corrections made. arXiv admin note: text overlap with arXiv:1203.6212

An exp-algebraic curve consists of a compact Riemann surface $S$ together with $n$ equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions, $\HH = \{ [h_1], \cdots, [h_n] \}$, with poles of orders $d_1, \cdots, d_n \geq 1$ at points $p_1, \cdots, p_n$. This data determines a space of functions $\OO_{\HH}$ (respectively, a space of $1$-forms $\Omega^0_{\HH}$) holomorphic on the punctured surface $S' = S - \{p_1, \cdots, p_n\}$ with exponential singularities at the points $p_1, \cdots, p_n$ of types $[h_1], \cdots, [h_n]$, i.e., near $p_i$ any $f \in \OO_{\HH}$ is of the form $f = ge^{h_i}$ for some germ of meromorphic function $g$ (respectively, any $\omega \in \Omega^0_{\HH}$ is of the form $\omega = \alpha e^{h_i}$ for some germ of meromorphic $1$-form). For any $\omega \in \Omega^0_{\HH}$ the completion of $S'$ with respect to the flat metric $|\omega|$ gives a space $S^* = S' \cup \RR$ obtained by adding a finite set $\RR$ of $\sum_i d_i$ points, and it is known that integration along curves produces a nondegenerate pairing of the relative homology $H_1(S^*, \RR ; \C)$ with the deRham cohomology group defined by $H^1_{dR}(S, \HH) := \Omega^0_{\HH}/d\OO_{\HH}$. There is a degree zero line bundle $L_{\HH}$ associated to an exp-algebraic curve, with a natural isomorphism between $\Omega^0_{\HH}$ and the space $W_{\HH}$ of meromorphic $L_{\HH}$-valued $1$-forms which are holomorphic on $S'$, so that $H_1(S^*, \RR ; \C)$ maps to a subspace $K_{\HH} \subset W^*_{\HH}$. We show that the exp-algebraic curve $(S, \HH)$ is determined uniquely by the pair $(L_{\HH},\, K_{\HH} \subset W^*_{\HH})$., Comment: arXiv admin note: substantial text overlap with arXiv:1602.08219

Let $f$ be a germ of holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in $\mathbb{C}$ (i.e. $f(0) = 0, f'(0) = e^{2\pi i \alpha}, \alpha \in \mathbb{R} - \mathbb{Q}$). Perez-Marco showed the existence of a unique continuous monotone one-parameter family of nontrivial invariant full continua containing the fixed point called Siegel compacta, and gave a correspondence between germs and families $(g_t)$ of circle maps obtained by conformally mapping the complement of these compacts to the complement of the unit disk. The family of circle maps $(g_t)$ is the orbit of a locally-defined semigroup $(\Phi_t)$ on the space of analytic circle maps which we show has a well-defined infinitesimal generator $X$. The explicit form of $X$ is obtained by using the Loewner equation associated to the family of hulls $(K_t)$. We show that the Loewner measures $(\mu_t)$ driving the equation are 2-conformal measures on the circle for the circle maps $(z \mapsto \overline{g_t(\overline{z})})$., Comment: 16 pages. Corrected typos, added section on linearizable maps and conformal radius

Log-Riemann surfaces of finite type are Riemann surfaces with finitely generated fundamental group equipped with a local diffeomorphism to C such that the surface has finitely many infinite order ramification points. We define and prove nondegeneracy of a period pairing for log-Riemann surfaces of finite type, given by pairing differentials with finitely many exponential singularities, of the form g exp(\int R_0) dz (where g, R_0 are meromorphic functions on a compact Riemann surface, with R_0 fixed) with closed curves and curves joining infinite order ramification points. As a consequence we show that the dimension of a cohomology group (given by differentials with exponential singularities of fixed type, modulo differentials of functions with exponential singularities of the same fixed type) is finite, equal to (2g + #R + (n-2)), where g is the genus of the compact Riemann surface, R is the set of infinite order ramification points, and n the number of exponential singularities.

For a generic class of rational functions, we give an explicit description of the flat structure on the Riemann sphere induced by a meromorphic 1-form R(z)dz, where R is a rational function. The rational functions in the generic class we consider have only simple poles. We show that the flat structure may be obtained by pasting isometrically flat half-cylinders to a 'log-polygon', which is a domain bounded by straight line segments in a simply connected finite sheeted branched cover of C.

We introduce the notion of log-Riemann surfaces. These are Riemann surfaces given by cutting and pasting planes together isometrically, and come equipped with a holomorphic local diffeomorphism to C called the projection map, and a corresponding flat metric obtained by pulling back the Euclidean metric. We define ramification points to be the points added in the metric completion of the surface with respect to the induced path metric; any such point has a well-defined order $1 \leq n \leq +\infty$ such that the projection map restricted to a small punctured neighbourhood of the point is an $n-to-1$ covering of a punctured disk in C. We prove that simply connected log-Riemann surfaces with finitely many ramification points are biholomorphic to C and the uniformization, with respect to the distinguished charts on the surface given by the projection map, is given by an entire function of the form $F(z) = \int Q(z)e^{P(z)} dz$ where $P, Q$ are polynomials of degrees equal to the number of infinite and finite order ramification points respectively. We also develop an algebraic theory for such log-Riemann surfaces, defining a ring of functions on the surface with finite values at all ramification points, such that the ring separates all points including the infinite order ramification points.

Let $(X, g_0)$ be a simply connected, complete, negatively curved Riemannian manifold. We prove local and infinitesimal rigidity results for compactly supported deformations of the metric $g_0$. For any negatively curved metric $g$ equal to $g_0$ outside a compact, the identity map of $X$ induces a natural boundary map between the boundaries at infinity of $X$ with respect to $g_0$ and $g$. We show that if $(g_t)$ is a smooth 1-parameter family of negatively curved metrics all equal to $g_0$ outside a fixed compact then if all the boundary maps (between the boundaries of $X$ with respect to $g_0$ and $g_t$) are Moebius then the metrics $g_t$ are all isometric to $g_0$. We also show that given a compact $K$ in $X$, there is a neighbourhood of $g_0$ in the $C^{2,\alpha}$ topology such that for any negatively curved metric $g$ in this neighbourhood which is equal to $g_0$ outside $K$, if the boundary map is Moebius and the $g_0$ and $g$ volumes of $K$ agree then $g$ is isometric to $g_0$.

We consider a log-Riemann surface $\mathcal{S}$ with a finite number of ramification points and finitely generated fundamental group. The log-Riemann surface is equipped with a local holomorphic difffeomorphism $\pi : \mathcal{S} \to \C$. We prove that $\mathcal{S}$ is biholomorphic to a compact Riemann surface with finitely many punctures $S$, and the pull-back of the 1-form $d\pi$ under the biholomorphic map $\phi : S \to \mathcal{S}$ is a 1-form $\omega = \phi^* d\pi$ with isolated singularities at the punctures of exponential type, i.e. near each puncture $p$, $\omega = e^h \cdot \omega_0$ where $h$ is a function meromorphic near $p$ and $\omega_0$ a 1-form meromorphic near $p$.

We consider Moebius and conformal homeomorphisms $f : \partial X \to \partial Y$ between boundaries of CAT(-1) spaces $X,Y$ equipped with visual metrics. A conformal map $f$ induces a topological conjugacy of the geodesic flows of $X$ and $Y$, which is flip-equivariant if $f$ is Moebius. We define a function $S(f) : \partial ^2 X \to \mathbb{R}$, the {\it integrated Schwarzian} of $f$, which measures the deviation of the topological conjugacy from being flip-equivariant, in particular vanishing if $f$ is Moebius. Conversely if $X,Y$ are simply connected complete manifolds with pinched negative sectional curvatures, then $f$ is Moebius on any open set $U \subset \partial X$ such that $S(f)$ vanishes on $\partial^2 U$. Indeed we obtain an explicit formula for the cross-ratio distortion in terms of the integrated Schwarzian. For such manifolds, we show that there is a Moebius homeomorphism $f : \partial X \to \partial Y$ if and only if there is a topological conjugacy of geodesic flows $\phi : T^1 X \to T^1 Y$ with a certain uniform continuity property along geodesics. We show that if $X,Y$ are proper, geodesically complete CAT(-1) spaces then any Moebius homeomorphism $f$ extends to a $(1, \log 2)$-quasi-isometry with image $\frac{1}{2}\log 2$-dense in $Y$. We prove that if $X,Y$ are in addition metric trees then $f$ extends to a surjective isometry. For $C^1$ conformal maps $f : \partial X \to \partial Y$ with bounded integrated Schwarzian and with domain $X$ a simply connected negatively curved manifold with a lower bound on sectional curvature, similar arguments show that $f$ extends to a $(1, \log 2 + 12||S(f)||_{\infty})$ quasi-isometry. We also obtain a dynamical classification of Moebius self-maps $f : \partial X \to \partial X$ into three types, elliptic, parabolic and hyperbolic., Comment: Added cross-ratio distortion formula for C1 conformal maps in terms of integrated Schwarzian, geometric mean value theorem and almost isometric extension of C1 conformal maps with bounded integrated Schwarzian

We consider simply connected log-Riemann surfaces with a finite number of ramification points. We prove that these surfaces are biholomorphic to C with uniformizations given by entire functions of the form F (z) = \int Q(z) e^{P(z)} dz where P, Q are polynomials of degrees equal to the number of infinite and finite order ramification points respectively. Conversely any such entire function defines a simply connected log-Riemann surface with finitely many ramification points., Comment: 13 pages

We define the notion of log-Riemann surfaces and Caratheodory convergence of log-Riemann surfaces. We prove a convergence theorem for uniformizations of simply connected log-Riemann surfaces converging in the Caratheodory topology. We obtain as a corollary a purely geometric proof of Euler's formula (1 + z/n)^n -> e^z .

Let f be a germ of holomorphic diffeomorphism with an irra- tionally indifferent fixed point at the origin in C (i.e. f(0) = 0, f'(0) = e 2pi i alpha, alpha in R - Q). Perez-Marco showed the existence of a unique family of nontrivial invariant full continua containing the fixed point called Siegel compacta. When f is non-linearizable (i.e. not holomorphically conjugate to the rigid rotation R_{alpha}(z) = e 2pi i z) the invariant compacts obtained are called hedgehogs. Perez-Marco developed techniques for the construction of examples of non-linearizable germs; these were used by the author to construct hedge- hogs of Hausdorff dimension one, and adapted by Cheritat to construct Siegel disks with pseudo-circle boundaries. We use these techniques to construct hedgehogs of positive area and hedgehogs with inaccessible fixed points., Comment: Typos and part of proof rewritten

Let $f(z) = e^{2\pi i \alpha}z + O(z^2), \alpha \in \mathbb{R}$ be a germ of holomorphic diffeomorphism in $\mathbb{C}$. For $\alpha$ rational and $f$ of infinite order, the space of conformal conjugacy classes of germs topologically conjugate to $f$ is parametrized by the Ecalle-Voronin invariants (and in particular is infinite-dimensional). When $\alpha$ is irrational and $f$ is nonlinearizable it is not known whether $f$ admits quasi-conformal deformations. We show that if $f$ has a sequence of repelling periodic orbits converging to the fixed point then $f$ embeds into an infinite-dimensional family of quasi-conformally conjugate germs no two of which are conformally conjugate.

Let f_1,...,f_N be commuting germs of holomorphic diffeomorphisms in C fixing the origin with irrational rationally independent rotation numbers alpha_1,...,alpha_N. We adapt Yoccoz' renormalization of germs to this setting to show that a Brjuno-type condition on simultaneous Diophantine approximability of the rotation numbers is sufficient for simultaneous linearizability of f_1,...,f_N. This generalizes a result of Moser's. In the absence of periodic orbits we show that a weaker arithmetic condition analogous to that of Perez-Marco's for the case of a single germ is sufficent for linearizability. We also obtain lower bounds for the conformal radii of the Siegel disks in both cases in terms of the arithmetic functions defining the arithmetic conditions., Comment: preliminary version

We present a construction of hedgehogs for holomorphic maps with an indifferent fixed point. We construct, for a family of commuting non-linearisable maps, a common hedgehog of Hausdorff dimension 1, the minimum possible., Comment: one figure omitted

Perez-Marco proved the existence of non-trivial totally invariant connected compacts called hedgehogs near the fixed point of a nonlinearizable germ of holomorphic diffeomorphism. We show that if two nonlinearisable holomorphic germs with a common indifferent fixed point have a common hedgehog then they must commute. This allows us to establish a correspondence between hedgehogs and nonlinearizable maximal abelian subgroups of Diff$(\bf C,0)$. We also show that two nonlinearizable germs are conjugate if and only if their rotation numbers are equal and a hedgehog of one can be mapped conformally onto a hedgehog of the other. Thus the conjugacy class of a nonlinearizable germ is completely determined by its rotation number and the conformal class of its hedgehogs., Comment: 11 pages

We study the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group $G_1$ and a quasiconformal conjugate $h^{-1}G_2 h$ of a cocompact group $G_2$. We show that if the conjugacy $h$ is not conformal then this group contains a non-trivial one parameter subgroup. This leads to rigidity results; for example, Mostow rigidity is an immediate consequence. We are also able to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group $G$, by a $G$-invariant pattern we mean a $G$-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of $G$. We prove that (in dimension at least three) for $G_1, G_2$ cocompact Kleinian groups, any quasiconformal map pairing a $G_1$-invariant pattern to a $G_2$-invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas-Mj who proved rigidity for Poincare Duality subgroups.

In the study of the local dynamics of a germ of diffeomorphism fixing the origin in C, an important problem is to determine the centralizer of the germ in the group Diff(C,0) of germs of diffeomorphisms fixing the origin. When the germ is not of finite order, then the centralizer is abelian, and hence a maximal abelian subgroup of Diff(C,0). Conversely any maximal abelian subgroup which contains an element of infinite order is equal to the centralizer of that element. A natural question is whether every maximal abelian subgroup contains an element of infinite order, or whether there exist maximal abelian torsion subgroups; we show that such subgroups do indeed exist, and moreover that any infinite subgroup of the rationals modulo the integers Q/Z can be embedded into Diff(C,0) as such a subgroup.

We use techniques of tube-log Riemann surfaces due to R.Perez-Marco to construct a hedgehog containing smooth $C^{\infty}$ combs. The hedgehog is a common hedgehog for a family of commuting non-linearisable holomorphic maps with a common indifferent fixed point. The comb is made up of smooth curves, and is transversally bi-H\"older regular.

For $i= 1,2$, let $G_i$ be cocompact groups of isometries of hyperbolic space $\Hyp^n$ of real dimension $n$, $n \geq 3$. Let $H_i \subset G_i$ be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of $H_i$ is a codimension one topological sphere. 2) limit set of $H_i$ is an even dimensional topological sphere. 3) $H_i$ is a codimension one duality group. This generalizes (1). In particular, if $n = 3$, $H_i$ could be any freely indecomposable subgroup of $G_i$. 4) $H_i$ is an odd-dimensional Poincare Duality group $PD(2k+1)$. This generalizes (2). We prove pattern rigidity for such pairs extending work of Schwartz who proved pattern rigidity when $H_i$ is cyclic. All this generalizes to quasiconvex subgroups of uniform lattices in rank one symmetric spaces satisfying one of the conditions (1)-(4), as well as certain special subgroups with disconnected limit sets. In particular, pattern rigidity holds for all quasiconvex subgroups of hyperbolic 3-manifolds that are not virtually free. Combining this with a result of Mosher-Sageev-Whyte, we get quasi-isometric rigidity results for graphs of groups where the vertex groups are uniform lattices in rank one symmetric spaces and edge groups are of any of the above types., Comment: v3: 23 pages, no figs, Final version incorporating referee's comments. To appear in "Groups, Geometry and Dynamics"