Hyperbolic geometry of cosmological attractors

Structural Stability and Hyperbolic Attractors

A necessary condition for structural stability is presented that in the two dimensional case means that the system has a finite number of topological attractors. Introduction. One of the basic questions in Dynamical System Theory is the characterization and the study of the properties of structurally stable systems (vector fields or diffeomorphisms). The standing conjecture about characterizati...

Hyperbolic Geometry

The explosion of singular hyperbolic attractors

A singular hyperbolic attractor for flows is a partially hyperbolic at-tractor with singularities (hyperbolic ones) and volume expanding central direction [MPP1]. The geometric Lorenz attractor [GW] is an example of a singular hyperbolic attractor. In this paper we study the perturbations of singular hyperbolic attractors for three-dimensional flows. It is proved that any attractor obtained fro...

Topological dimension of singular - hyperbolic attractors

An attractor is a transitive set of a flow to which all positive orbit close to it converges. An attractor is singular-hyperbolic if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding central direction [16]. The geometric Lorenz attractor [6] is an example of a singular-hyperbolic attractor with topological dimension ≥ 2. We shall prove that all singular-hyp...

Hyperbolic Geometry: Isometry Groups of Hyperbolic Space

The goal of this paper is twofold. First, it consists of an introduction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that preserve orientation, and deals with the topological properties of their discrete subgroups. In the...