ON THE NUMBER OF ERGODIC PHYSICAL/SRB MEASURES OF SINGULAR-HYPERBOLIC ATTRACTING SETS

The paper’s Theorem A states and proves that for a connected singular-hyperbolic attracting set of a C2 three-dimensional flow, the number s of ergodic physical measures whose support contains a singularity satisfies s ≤ 2·sL, where sL counts Lorenz-like equilibria; the proof hinges on Lemma 3.10: at most two distinct SRB measures can accumulate the same Lorenz-like singularity, using absolute continuity of the stable foliation and a two-sided geometry at the singularity, and then summing over singularities yields the bound. This argument is explicitly given and completed in Section 3.3.3, with the statement of Theorem A and examples showing sharpness (s = 2·sL) also constructed in the paper. The candidate (model) solution proves the same bound by reducing to a global Poincaré return map, quotienting along strong-stable leaves to a piecewise-expanding interval map and then invoking a basin-counting argument near the discontinuity associated to each Lorenz-like singularity. This yields the same conclusion via different machinery. A few technical steps in the model’s outline (global section/quotient regularity, integrability of the roof, and the claim that a whole one-sided neighborhood near the discontinuity lies in the basin) are standard but not fully justified in the outline; still, under customary hypotheses from the cited literature, the model’s proof is sound and aligns with the paper’s result. Overall: both correct, with different proofs, the paper’s being self-contained in the provided framework.