An Ergodic Spectral Decomposition Theorem for Singular Star Flows

TL;DR

This paper proves that for a broad class of singular star flows, there are finitely many ergodic measures of maximal entropy and equilibrium states, extending spectral decomposition concepts to flows with singularities.

Contribution

It establishes an ergodic spectral decomposition theorem for singular star flows, showing finiteness of measures and stability properties in the $C^1$ topology.

Findings

Finiteness of ergodic measures of maximal entropy for singular star flows.

Finiteness of equilibrium states for certain H"older potentials.

Almost expansiveness and continuous pressure variation in $C^1$ topology.

Abstract

For Axiom A diffeomorphisms and flows, the celebrated Spectral Decomposition Theorem of Smale states that the non-wandering set decomposes into a finite disjoint union of isolated compact invariant sets, each of which is the homoclinic class of a periodic orbit. For singular star flows which can be seen as ``Axiom A flows with singularities'', this result remains open and is known as the Spectral Decomposition Conjecture. In this paper, we will provide a positive answer to an ergodic version of this conjecture: $C^1$ open and densely, singular star flows with positive topological entropy can only have finitely many ergodic measures of maximal entropy. More generally, we obtain the finiteness of equilibrium states for any H\"older continuous potential functions satisfying a mild, yet optimal, condition. We also show that $C^1$ open and densely, star flows are almost expansive, and the topological pressure of continuous functions varies continuously with respect to the vector field in $C^1$ topology.