In this paper we show that there is an analytic and computable dynamical system with a hyperbolic sink s such that the basin of attraction of s is not computable. As a consequence, the problem of determining basins of attraction for analytic hyperbolic systems is not, in general, algorithmically solvable. 1 Summary of the paper Dynamical systems are fascinating mathematical objects. They can be...

We implement a semi-analytic scheme for numerically computing high order polynomial approximations of the stable and unstable manifolds associated with the fixed points of the normal form for the family of quadratic volume-preserving diffeomorphisms with quadratic inverse. We use this numerical scheme to study some hyperbolic dynamics associated with an invariant structure called a vortex-bubbl...

6 Hyperbolic Geometry 31 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.2 Ideal Tetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.3 Volume of Ideal Hyperbolic Tetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.4 Hyperbolic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . ....

A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a ...

The purpose of this document is to list some outstanding questions in the subject of hyperbolic geometry (real, complex, quaternionic, and the Cayley plane), with the primary focus on the latter three. This list was inspired by the problem sessions held at the conference on Complex Hyperbolic Geometry in Luminy (CIRM) in July of 2003. The questions are roughly grouped by similarity. The name(s)...

We implement a semi-analytic scheme for numerically computing high order polynomial approximations of the stable and unstable manifolds associated with the fixed points of the normal form for the family of quadratic volume-preserving diffeomorphisms with quadratic inverse. We use this numerical scheme to study some hyperbolic dynamics associated with an invariant structure called a vortex bubbl...

There are two types of parallel lines in Hyperbolic Geometry. There are those who diverge from each other in both directions (type 1) and those that diverge in one direction but come arbitrarily close to each other in the other direction (type 2). The first type have a minimum positive distance at points on a common perpendicular. The second type have no minimum distance: the distance tends to ...