An entropy dichotomy for singular star flows

TL;DR

This paper establishes a dichotomy for generic $C^1$ star flows, showing that non-trivial chain recurrent classes either have zero entropy or are isolated, with implications for their hyperbolic structure and stability.

Contribution

It introduces a new entropy dichotomy for singular star flows and characterizes the structure and stability of their chain recurrent classes.

Findings

Non-trivial chain recurrent classes are either zero entropy or isolated.

Zero entropy classes are sectional hyperbolic and undetectable by non-trivial ergodic measures.

Generic star flows have finitely many Lyapunov stable chain recurrent classes.

Abstract

We show that non-trivial chain recurrent classes for generic $C^1$ star flows satisfy a dichotomy: either they have zero topological entropy, or they must be isolated. Moreover, chain recurrent classes for generic star flows with zero entropy must be sectional hyperbolic, and cannot be detected by any non-trivial ergodic invariant probability. As a result, we show that $C^1$ generic star flows have only finitely many Lyapunov stable chain recurrent classes.