221 research outputs found
Euclidean and hyperbolic lenghs of images of arcs
Let be a function that is analytic in the unit disc. We give new
estimates, and new proofs of existing estimates, of the Euclidean length of the
image under of a radial segment in the unit disc. Our methods are based on
the hyperbolic geometry of plane domains, and we address some new questions
that follow naturally from this approach.Comment: 26 page
Electrical networks on -simplex fractals
The decimation map for a network of admittances on an
-simplex lattice fractal is studied. The asymptotic behaviour of
for large-size fractals is examined. It is found that in the
vicinity of the isotropic point the eigenspaces of the linearized map are
always three for ; they are given a characterization in terms of
graph theory. A new anisotropy exponent, related to the third eigenspace, is
found, with a value crossing over from to
.Comment: 14 pages, 8 figure
On the Plants Leaves Boundary, "Jupe \`a Godets" and Conformal Embeddings
The stable profile of the boundary of a plant's leaf fluctuating in the
direction transversal to the leaf's surface is described in the framework of a
model called a "surface \`a godets". It is shown that the information on the
profile is encoded in the Jacobian of a conformal mapping (the coefficient of
deformation) corresponding to an isometric embedding of a uniform Cayley tree
into the 3D Euclidean space. The geometric characteristics of the leaf's
boundary (like the perimeter and the height) are calculated. In addition a
symbolic language allowing to investigate statistical properties of a "surface
\`a godets" with annealed random defects of curvature of density is
developed. It is found that at the surface exhibits a phase transition
with critical exponent from the exponentially growing to the flat
structure.Comment: 17 pages (revtex), 8 eps-figures, to appear in Journal of Physics
Kick stability in groups and dynamical systems
We consider a general construction of ``kicked systems''. Let G be a group of
measure preserving transformations of a probability space. Given its
one-parameter/cyclic subgroup (the flow), and any sequence of elements (the
kicks) we define the kicked dynamics on the space by alternately flowing with
given period, then applying a kick. Our main finding is the following stability
phenomenon: the kicked system often inherits recurrence properties of the
original flow. We present three main examples. 1) G is the torus. We show that
for generic linear flows, and any sequence of kicks, the trajectories of the
kicked system are uniformly distributed for almost all periods. 2) G is a
discrete subgroup of PSL(2,R) acting on the unit tangent bundle of a Riemann
surface. The flow is generated by a single element of G, and we take any
bounded sequence of elements of G as our kicks. We prove that the kicked system
is mixing for all sufficiently large periods if and only if the generator is of
infinite order and is not conjugate to its inverse in G. 3) G is the group of
Hamiltonian diffeomorphisms of a closed symplectic manifold. We assume that the
flow is rapidly growing in the sense of Hofer's norm, and the kicks are
bounded. We prove that for a positive proportion of the periods the kicked
system inherits a kind of energy conservation law and is thus superrecurrent.
We use tools of geometric group theory and symplectic topology.Comment: Latex, 40 pages, revised versio
Periodic boundary conditions on the pseudosphere
We provide a framework to build periodic boundary conditions on the
pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian
space of constant negative curvature. Starting from the common case of periodic
boundary conditions in the Euclidean plane, we introduce all the needed
mathematical notions and sketch a classification of periodic boundary
conditions on the hyperbolic plane. We stress the possible applications in
statistical mechanics for studying the bulk behavior of physical systems and we
illustrate how to implement such periodic boundary conditions in two examples,
the dynamics of particles on the pseudosphere and the study of classical spins
on hyperbolic lattices.Comment: 30 pages, minor corrections, accepted to J. Phys.
On the growth of nonuniform lattices in pinched negatively curved manifolds
We study the relation between the exponential growth rate of volume in a
pinched negatively curved manifold and the critical exponent of its lattices.
These objects have a long and interesting story and are closely related to the
geometry and the dynamical properties of the geodesic flow of the manifold
The orbifold transform and its applications
We discuss the notion of the orbifold transform, and illustrate it on simple
examples. The basic properties of the transform are presented, including
transitivity and the exponential formula for symmetric products. The connection
with the theory of permutation orbifolds is addressed, and the general results
illustrated on the example of torus partition functions
Topological Lensing in Spherical Spaces
This article gives the construction and complete classification of all
three-dimensional spherical manifolds, and orders them by decreasing volume, in
the context of multiconnected universe models with positive spatial curvature.
It discusses which spherical topologies are likely to be detectable by
crystallographic methods using three-dimensional catalogs of cosmic objects.
The expected form of the pair separation histogram is predicted (including the
location and height of the spikes) and is compared to computer simulations,
showing that this method is stable with respect to observational uncertainties
and is well suited for detecting spherical topologies.Comment: 32 pages, 26 figure
Harmonic analysis of iterated function systems with overlap
In this paper we extend previous work on IFSs without overlap. Our method
involves systems of operators generalizing the more familiar Cuntz relations
from operator algebra theory, and from subband filter operators in signal
processing.Comment: 37 page
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