12,293 research outputs found
Isotopic tiling theory for hyperbolic surfaces
In this paper, we develop the mathematical tools needed to explore isotopy
classes of tilings on hyperbolic surfaces of finite genus, possibly
nonorientable, with boundary, and punctured. More specifically, we generalize
results on Delaney-Dress combinatorial tiling theory using an extension of
mapping class groups to orbifolds, in turn using this to study tilings of
covering spaces of orbifolds. Moreover, we study finite subgroups of these
mapping class groups. Our results can be used to extend the Delaney-Dress
combinatorial encoding of a tiling to yield a finite symbol encoding the
complexity of an isotopy class of tilings. The results of this paper provide
the basis for a complete and unambiguous enumeration of isotopically distinct
tilings of hyperbolic surfaces
Genus 0 characteristic numbers of the tropical projective plane
Finding the so-called characteristic numbers of the complex projective plane
is a classical problem of enumerative geometry posed by
Zeuthen more than a century ago. For a given and one has to find the
number of degree genus curves that pass through a certain generic
configuration of points and at the same time are tangent to a certain generic
configuration of lines. The total number of points and lines in these two
configurations is so that the answer is a finite integer number.
In this paper we translate this classical problem to the corresponding
enumerative problem of tropical geometry in the case when . Namely, we
show that the tropical problem is well-posed and establish a special case of
the correspondence theorem that ensures that the corresponding tropical and
classical numbers coincide. Then we use the floor diagram calculus to reduce
the problem to pure combinatorics. As a consequence, we express genus 0
characteristic numbers of \CC P^2 in terms of open Hurwitz numbers.Comment: 55 pages, 23 figure
Generalised shear coordinates on the moduli spaces of three-dimensional spacetimes
We introduce coordinates on the moduli spaces of maximal globally hyperbolic
constant curvature 3d spacetimes with cusped Cauchy surfaces S. They are
derived from the parametrisation of the moduli spaces by the bundle of measured
geodesic laminations over Teichm\"uller space of S and can be viewed as
analytic continuations of the shear coordinates on Teichm\"uller space. In
terms of these coordinates the gravitational symplectic structure takes a
particularly simple form, which resembles the Weil-Petersson symplectic
structure in shear coordinates, and is closely related to the cotangent bundle
of Teichm\"uller space. We then consider the mapping class group action on the
moduli spaces and show that it preserves the gravitational symplectic
structure. This defines three distinct mapping class group actions on the
cotangent bundle of Teichm\"uller space, corresponding to different values of
the curvature.Comment: 40 pages, 6 figure
Symbolic extensions in intermediate smoothness on surfaces
We prove that maps with on a compact surface have
symbolic extensions, i.e. topological extensions which are subshifts over a
finite alphabet. More precisely we give a sharp upper bound on the so-called
symbolic extension entropy, which is the infimum of the topological entropies
of all the symbolic extensions. This answers positively a conjecture of
S.Newhouse and T.Downarowicz in dimension two and improves a previous result of
the author \cite{burinv}.Comment: 27 page
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