1,757 research outputs found
Hyperbolicity on Graph Operators
A graph operator is a mapping F : Gamma → Gamma 0 , where Gamma and Gamma 0 are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Lambda(G); subdivision graph, S(G); total graph, T(G); and the operators R(G) and Q(G). In particular, we get relationships such as delta(G) ≤ delta(R(G)) ≤ delta(G) + 1/2, delta(Lambda(G)) ≤ delta(Q(G)) ≤ delta(Lambda(G)) + 1/2, delta(S(G)) ≤ 2delta(R(G)) ≤ delta(S(G)) + 1 and delta(R(G)) − 1/2 ≤ delta(Lambda(G)) ≤ 5delta(R(G)) + 5/2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.Supported in part by two grants from Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain
On the Hyperbolicity of Lorenz Renormalization
We consider infinitely renormalizable Lorenz maps with real critical exponent
and combinatorial type which is monotone and satisfies a long return
condition. For these combinatorial types we prove the existence of periodic
points of the renormalization operator, and that each map in the limit set of
renormalization has an associated unstable manifold. An unstable manifold
defines a family of Lorenz maps and we prove that each infinitely
renormalizable combinatorial type (satisfying the above conditions) has a
unique representative within such a family. We also prove that each infinitely
renormalizable map has no wandering intervals and that the closure of the
forward orbits of its critical values is a Cantor attractor of measure zero.Comment: 63 pages; 10 figure
Singularities and Quantum Gravity
Although there is general agreement that a removal of classical gravitational
singularities is not only a crucial conceptual test of any approach to quantum
gravity but also a prerequisite for any fundamental theory, the precise
criteria for non-singular behavior are often unclear or controversial. Often,
only special types of singularities such as the curvature singularities found
in isotropic cosmological models are discussed and it is far from clear what
this implies for the very general singularities that arise according to the
singularity theorems of general relativity. In these lectures we present an
overview of the current status of singularities in classical and quantum
gravity, starting with a review and interpretation of the classical singularity
theorems. This suggests possible routes for quantum gravity to evade the
devastating conclusion of the theorems by different means, including modified
dynamics or modified geometrical structures underlying quantum gravity. The
latter is most clearly present in canonical quantizations which are discussed
in more detail. Finally, the results are used to propose a general scheme of
singularity removal, quantum hyperbolicity, to show cases where it is realized
and to derive intuitive semiclassical pictures of cosmological bounces.Comment: 41 pages, lecture course at the XIIth Brazilian School on Cosmology
and Gravitation, September 200
Global hyperbolicity of renormalization for C^r unimodal mappings
In this paper we extend M. Lyubich's recent results on the global
hyperbolicity of renormalization of quadratic-like germs to the space of C^r
unimodal maps with quadratic critical point. We show that in this space the
bounded-type limit sets of the renormalization operator have an invariant
hyperbolic structure provided r ge 2+ alpha with alpha close to one. As an
intermediate step between Lyubich's results and ours, we prove that the
renormalization operator is hyperbolic in a Banach space of real analytic maps.
We construct the local stable manifolds and prove that they form a continuous
lamination whose leaves are C^1 codimension one, Banach submanifolds of the
ambient space, and whose holonomy is C^{1+\beta} for some beta >0. We also
prove that the global stable sets are C^1 immersed (codimension one)
submanifolds as well, provided r ge 3+ alpha with alpha close to one. As a
corollary, we deduce that in generic, one-parameter families of C^r unimodal
maps, the set of parameters corresponding to infinitely renormalizable maps of
bounded combinatorial type is a Cantor set with Hausdorff dimension less than
one.Comment: 94 pages, published versio
Algorithms for computing normally hyperbolic invariant manifolds
An effcient algorithm is developed for the numerical computation of normally hyperbolic invariant manifolds, based on the graph transform and Newton's method. It fits in the perturbation theory of discrete dynamical systems and therefore allows application to the setting of continuation. A convergence proof is included. The scope of application is not restricted to hyperbolic attractors, but extends to normally hyperbolic manifolds of saddle type. It also computes stable and unstable manifolds. The method is robust and needs only little specification of the dynamics, which makes it applicable to e.g. Poincaré maps. Its performance is illustrated on examples in 2D and 3D, where a numerical discussion is included.
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