Topologically stable manifolds for index-1 singular dominated splittings

2025/05/11 by Crovisier, Sylvain, Yang, Dawei
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.06942

Abstract

For C² vector fields, we study regular ergodic measures whose supports admit singular dominated splittings with one of the bundles having dimension 1. For such a measure μ, we prove that if any periodic orbit within the support of μ (when it exists) has at least one negative Lyapunov exponent, and if the dynamics on the support of μ is not topologically equivalent to an irrational flow on a 2-torus, then μ-almost every point x admits a 2-dimensional topologically stable manifold V^s(x): we mean that V^s(x) is an embedded disc such that the orbit any point within it converges to the orbit of x up to a time-reparametrization. Note that we do not assume any hyperbolicity for μ. We also establish an analogous conclusion for compact invariant sets Λ with a singular dominated splitting, assuming some mild contraction property (any regular ergodic measure properly supported in Λ must have at least one negative Lyapunov exponent). This result will be used in our future work on the Palis density conjecture for three-dimensional vector fields.

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