An Ergodic Spectral Decomposition Theorem for Singular Star Flows

Abstract

For Axiom A diffeomorphisms and flows, Smale's Spectral Decomposition Theorem asserts that the non-wandering set decomposes into finitely many isolated hyperbolic basic sets, each given by a homoclinic class. For singular star flows, which may be viewed as "Axiom A flows with singularities", the corresponding spectral decomposition remains open and is known as the Spectral Decomposition Conjecture. We provide a positive answer to an ergodic formulation of this conjecture: $C^1$-open and dense among singular star flows with positive topological entropy, there is a unique measure of maximal entropy. More generally, we prove the uniqueness of equilibrium states for Hölder continuous potentials under a mild and natural pressure gap condition. We further establish that $C^1$-open and dense star flows are almost expansive and that the topological pressure of continuous potentials varies continuously with respect to the vector field in the $C^1$ topology. Our approach combines ergodic and geometric arguments adapted to the multi-singular setting. In this context, classical hyperbolic tools such as uniform local product structure or invariant splittings on the tangent bundle are no longer available. To overcome this, we develop new mechanisms to control the geometry of orbit segments and to produce transversal intersections on large subsets uniformly detected by good invariant measures. These ingredients allow us to extend classical arguments to the multi-singular setting through structural properties of equilibrium states combined with refined shadowing and specification at the level of invariant measures.

114 pages, 2 figures