arXiv subjects

Kingshook Biswas

Publications and source records attributed to Kingshook Biswas.

35 records · Page 2Linked to original sources

arXiv2016

Loewner evolution of hedgehogs and 2-conformal measures of circle maps

Let $f$ be a germ of holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in $\mathbb{C}$ (i.e. $f(0) = 0, f'(0) = e^{2πi α}, α\in \mathbb{R} - \mathbb{Q}$). 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 $(Φ_t)$ 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 $(μ_t)$ driving the equation are 2-conformal measures on the circle for the circle maps $(z \mapsto \overline{g_t(\overline{z})})$.

math.DS↗

arXiv2015

On tube-log Riemann surfaces and primitives of rational functions

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.

math.CV↗

arXiv2015

Log-Riemann Surfaces

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.

math.CV↗

arXiv2015

Uniformization of higher genus finite type log-Riemann surfaces

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 $π: \mathcal{S} \to \C$. 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 $dπ$ under the biholomorphic map $ϕ: S \to \mathcal{S}$ is a 1-form $ω= ϕ^* dπ$ with isolated singularities at the punctures of exponential type, i.e. near each puncture $p$, $ω= e^h \cdot ω_0$ where $h$ is a function meromorphic near $p$ and $ω_0$ a 1-form meromorphic near $p$.

math.CV↗

arXiv2015

Local and infinitesimal rigidity of simply connected negatively curved manifols

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 $C^{2,α}$ 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 $g_0$.

math.DG↗

arXiv2013

On Moebius and conformal maps between boundaries of CAT(-1) spaces

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 $ϕ: T^1 X \to T^1 Y$ 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.

math.DS↗

arXiv2013

Positive area and inaccessible fixed points for hedgehogs

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.

math.CV↗

arXiv2010

Caratheodory convergence of log-Riemann surfaces and Euler's formula

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 .

math.CV↗

arXiv2010

Uniformization of simply connected finite type log-Riemann surfaces

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.

math.CV↗

arXiv2010

Pattern Rigidity in Hyperbolic Spaces: Duality and PD Subgroups

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.

math.GT↗

arXiv2010

Quasi-conformal deformations of nonlinearizable germs

Let $f(z) = e^{2πi α}z + O(z^2), α\in \mathbb{R}$ be a germ of holomorphic diffeomorphism in $\mathbb{C}$. For $α$ 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 $α$ 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.

math.CV↗

arXiv2009

Simultaneous linearization of commuting germs of holomorphic diffeomorphisms

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.

math.DS↗

arXiv2009

Hedgehogs of Hausdorff dimension one

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.

math.DS↗

arXiv2009

Complete Conjugacy Invariants of Nonlinearizable Holomorphic Dynamics

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.

math.DS↗

arXiv2009

Maximal Abelian Torsion Subgroups of Diff(C,0)

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.

math.DS↗

arXiv2009

Smooth Combs Inside Hedgehogs

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-Hölder regular.

math.DS↗

arXiv2009

Flows, Fixed Points and Rigidity for Kleinian Groups

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.

math.GT↗