530 research outputs found
Lattices and Their Continuum Limits
We address the problem of the continuum limit for a system of Hausdorff
lattices (namely lattices of isolated points) approximating a topological space
. The correct framework is that of projective systems. The projective limit
is a universal space from which can be recovered as a quotient. We dualize
the construction to approximate the algebra of continuous
functions on . In a companion paper we shall extend this analysis to systems
of noncommutative lattices (non Hausdorff lattices).Comment: 11 pages, 1 Figure included in the LaTeX Source New version, minor
modifications (typos corrected) and a correction in the list of author
Discrete-Euclidean operations
International audienceIn this paper we study the relationship between the Euclidean and the discrete space. We study discrete operations based on Euclidean functions: the discrete smooth scaling and the discrete-continuous rotation. Conversely, we study Euclidean oper- ations based on discrete functions: the discrete based simplification, the Euclidean- discrete union and the Euclidean-discrete co-refinement. These operations operate partly in the discrete and partly in the continuous space. Especially for the discrete smooth scaling operation, we provide error bounds when different such operations are chained
Distances on a Lattice from Non-commutative Geometry
Using the tools of noncommutative geometry we calculate the distances between
the points of a lattice on which the usual discretized Dirac operator has been
defined. We find that these distances do not have the expected behaviour,
revealing that from the metric point of view the lattice does not look at all
as a set of points sitting on the continuum manifold. We thus have an
additional criterion for the choice of the discretization of the Dirac
operator.Comment: 14 page
Spatial discretizations of generic dynamical systems
How is it possible to read the dynamical properties (ie when the time goes to
infinity) of a system on numerical simulations? To try to answer this question,
we study in this manuscript a model reflecting what happens when the orbits of
a discrete time system (for example an homeomorphism) are computed
numerically . The computer working in finite numerical precision, it will
replace by a spacial discretization of , denoted by (where the
order of discretization stands for the numerical accuracy). In particular,
we will be interested in the dynamical behaviour of the finite maps for a
generic system and going to infinity, where generic will be taken in
the sense of Baire (mainly among sets of homeomorphisms or
-diffeomorphisms).
The first part of this manuscript is devoted to the study of the dynamics of
the discretizations , when is a generic conservative/dissipative
homeomorphism of a compact manifold. We show that it would be mistaken to try
to recover the dynamics of from that of a single discretization : its
dynamics strongly depends on the order . To detect some dynamical features
of , we have to consider all the discretizations when goes through
.
The second part deals with the linear case, which plays an important role in
the study of -generic diffeomorphisms, discussed in the third part of this
manuscript. Under these assumptions, we obtain results similar to those
established in the first part, though weaker and harder to prove.Comment: 322 pages. This is an improved version of the thesis of the author
(among others, the introduction and conclusion have been translated into
English). In particular, it contains works already published on arXiv.
Comments welcome
A non-perturbative Lorentzian path integral for gravity
A well-defined regularized path integral for Lorentzian quantum gravity in
three and four dimensions is constructed, given in terms of a sum over
dynamically triangulated causal space-times. Each Lorentzian geometry and its
associated action have a unique Wick rotation to the Euclidean sector. All
space-time histories possess a distinguished notion of a discrete proper time.
For finite lattice volume, the associated transfer matrix is self-adjoint and
bounded. The reflection positivity of the model ensures the existence of a
well-defined Hamiltonian. The degenerate geometric phases found previously in
dynamically triangulated Euclidean gravity are not present. The phase structure
of the new Lorentzian quantum gravity model can be readily investigated by both
analytic and numerical methods.Comment: 11 pages, LaTeX, improved discussion of reflection positivity,
conclusions unchanged, references update
From the coarse geometry of warped cones to the measured coupling of groups
In this article, we prove that if two warped cones corresponding to two
groups with free, isometric, measure-preserving, ergodic actions on two
manifolds are quasi-isometric, then the corresponding groups are uniformly
measured equivalent (UME). It was earlier known from the work of de Laat-Vigolo
that if two warped cones are QI, then their stable products are QI. Our result
strengthens this result and go further to prove UME of the groups. However,
Fisher-Nguyen-Limbeek proves that if the warped cones corresponding to two
finitely presented groups with no free abelian factors are QI, then there is an
affine commensuration of the two actions. Our result can be seen as an
extension of their result in the setting of infinite presentability under some
extra assumptions.Comment: Comments welcome. arXiv admin note: text overlap with
arXiv:1512.08828. Some typos have been corrected and some minor gaps of
Proposition 4.2 have been filled u
- …