A proof of 𝐶¹ stability conjecture for three-dimensional flows

Hu, Sen

doi:10.1090/S0002-9947-1994-1172297-6

A proof of $C^ 1$ stability conjecture for three-dimensional flows
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Trans. Amer. Math. Soc. 342 (1994), 753-772 Request permission

Abstract:

We give a proof of the ${C^1}$ stability conjecture for three-dimensional flows, i.e., prove that there exists a hyperbolic structure over the $\Omega$ set for the structurally stable three-dimensional flows. Mañé’s proof for the discrete case motivates our proof and we find his perturbation techniques crucial. In proving this conjecture we have overcome several new difficulties, e.g., the change of period after perturbation, the ergodic closing lemma for flows, the existence of dominated splitting over $\Omega \backslash \mathcal {P}$ where $\mathcal {P}$ is the set of singularities for the flow, the discontinuity of the contracting rate function on singularities, etc. Based on these we finally succeed in separating the singularities from the other periodic orbits for the structurally stable systems, i.e., we create unstable saddle connections if there are accumulations of periodic orbits on the singularities.

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