Specification and partial hyperbolicity for flows

TL;DR

This paper investigates the conditions under which certain dynamical flows, specifically those with partially hyperbolic attractors and Lorenz attractors, fail to satisfy the specification property, highlighting structural constraints.

Contribution

It establishes that flows with specific hyperbolic structures and multiple saddles of different indices do not satisfy the specification property, including all Geometric Lorenz attractors.

Findings

Flows with partially hyperbolic attractors and multiple saddles lack the specification property.

All Geometric Lorenz attractors do not satisfy the specification property.

The structure of stable and strongly stable bundles influences the specification property.

Abstract

In this article we prove that if a flow exhibits a partially hyperbolic attractor and it has two periodic saddles with different indices, and the stable index of one of them coincides with the dimension of strongly stable bundles, then it does not satisfy the specification property. As an application, we prove that all Geometric Lorenz attractors do not satisfy the specification property.