Robust Transitivity and Singular-Hyperbolicity

Abstract

The spectral decomposition theorem, presented in Sect. 2.3 and first proved in (Smale, Bull. Am. Math. Soc. 73:747–817, 1967), provides a description of the non-wandering set of a structural stable system as a finite number of disjoint compact maximal invariant and transitive sets, each of these pieces being well understood from both the deterministic and from statistical viewpoints. Moreover such a decomposition persists under small C ¹ perturbations. This naturally leads to the study of isolated transitive sets that remain transitive for all nearby systems (robustness).

However these ideas were developed in the setting of uniform hyperbolicity. The Lorenz equations (2.2) provide an example of a robust attractor containing an equilibrium point at the origin and periodic points accumulating on it. This is a non-uniformly hyperbolic attractor which cannot be destroyed by any small perturbation of the parameters, i.e., a robustly transitive set but not hyperbolic.