The Lyapunov Exponents of Hyperbolic Measures for C^1 Star Vector Fields on Three-dimensional Manifolds

The PDF proves two statements for C^1 star vector fields on compact 3-manifolds: (A) periodic measures are dense among hyperbolic ergodic measures not supported on singularities, and (B) the two ψ*-Lyapunov exponents of such a measure are approximated by those of periodic orbits. The paper constructs long (η,T)-ψ* quasi-hyperbolic strings inside a positive-measure Pesin block, closes them via a shadowing lemma with a small time reparametrization, and controls the ψ*-distortion to pass to measures and Lyapunov exponents (Theorem A/B as stated and proved; see the statements and proofs around Theorem A and Theorem B, including the shadowing lemma Theorem 3.5 and its use in Propositions 5.1–5.2 ). The candidate solution follows the same architecture: Oseledets splitting on the normal bundle (two 1D subbundles), selection of quasi-hyperbolic orbit segments (via Pliss/Liao-type sifting), closing by a flow shadowing lemma formulated for ψ*, then (i) weak* approximation via closeness of tracks and small reparametrization, and (ii) exponent approximation using one-dimensionality and distortion control. The paper’s quantitative steps (e.g., the transfer-map bound |log∥T_i∥|<ε/2 and time-change bounds |θ'(t)−1|≤ε) match the model’s use of exponentially small ψ*-distortion and small time-change (e.g., the bounds derived after Theorem 3.5 and in the proofs of Propositions 5.1–5.2 ). Minor presentation differences aside, both arguments are the same in substance.