Topological attractors of contracting Lorenz maps

Abstract

We study the non-wandering set of contracting Lorenz maps. We show that if such a map f doesn't have any attracting periodic orbit, then there is a unique topological attractor. Precisely, there is a compact set Λ such that ω f (x) = Λ for a residual set of points x ∈ [0, 1]. Furthermore, we classify the possible kinds of attractors that may occur.Date: August 28, 2018. Partially supported by FAPERJ, CNPq and CAPES.

Hyperbolicity of renormalization for dissipative gap mappings

“…It would be difficult to cite all the papers studying this famous dynamical system, but for example see papers [1,3,15,18,39,49]. The success of the use of the one-dimensional Lorenz mapping in studying the flow has led to an extensive study of these interval mappings, see papers [6,14,19,20,26,29,30,43,50] among many others. Great progress in understanding the Cherry flow on a two-torus has followed from a similar approach [2,8,11,28,[33][34][35][36][37][38]40].…”

Hyperbolicity of renormalization for dissipative gap mappings

“…It would be difficult to cite all the papers studying this famous dynamical system, but for example see [1,49,18,3,15,39]. The success of the use of the one-dimensional Lorenz mapping in studying the flow has led to an extensive study of these interval mappings, see [43,19,28,20,50,14,30,26,6] among many others. Great progress in understanding the Cherry flow on a two-torus has followed from a similar approach [8,29,33,2,11,34,40,35,36,37,38].…”