2011.04824
Attractors of direct products
Stanislav Minkov, Ivan Shilin
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper explicitly constructs a C∞ flow on an annulus with a global physical (hence natural) SRB measure µϕ and shows that the square flow Φ also has a global SRB measure µΦ but µΦ ≠ µϕ×µϕ, with Amin(Φ) strictly smaller than Amin(ϕ)×Amin(ϕ) (Theorem 22) . The construction is based on “oscillating measures,” and the plan-of-proof spells out µϕ = 1/2(δL+δR) and µΦ = 1/2(δ(L,L)+δ(R,R)) , using the SRB definitions recalled in Section 2 . The candidate solution’s Poincaré–Bendixson-based exclusion misses this oscillatory-time-average mechanism and incorrectly asserts that a global SRB on an annulus forces either a single sink or a single attracting limit cycle; the paper’s example contradicts that claim.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper presents clear counterexamples showing that, for several notions of attractor (Milnor, statistical, minimal) and for SRB supports, the attractor of a square flow can be strictly smaller than the square of the attractor. The annulus construction with oscillating measures producing global physical/natural SRB measures for both the flow and its square is novel and addresses a natural multiplicativity question. Some steps in Section 7 are outlined at a high level; adding a few local estimates and a more explicit choice of functions ensuring smoothness/flatness would further strengthen the exposition.