The Specification Property for Flows from the Robust and Generic Viewpoint

TL;DR

The paper demonstrates that robust weak specification implies hyperbolic, mixing, and Anosov properties for flows, establishing a strong link between the specification property and hyperbolic dynamics.

Contribution

It proves that robust weak specification leads to hyperbolic and mixing behavior, and that generic flows with this property are Anosov, connecting specification with hyperbolic dynamics.

Findings

Robust weak specification implies hyperbolic, topologically mixing sets.

Flows with robust weak specification are Anosov flows.

Generic flows with weak specification are Anosov.

Abstract

We prove that if $X|_\Lambda$ has the weak specification property robustly, where $\Lambda$ is an isolated set, then $\Lambda$ is a hyperbolic topologically mixing set and, as a consequence, if $X$ is a vector field that has the weak specification property robustly on a closed manifold $M$, then the flow $X_t$ is a topologically mixing Anosov flow. Also we prove that there exists a residual subset $\SR \in \Mundo$ so that if $X \in \SR$ and $X$ has the weak specification property, then $X_t$ is an Anosov flow.