2006.04293
SMOOTH MIXING ANOSOV FLOWS IN DIMENSION THREE ARE EXPONENTIAL MIXING
Masato Tsujii, Zhiyuan Zhang
correctmedium confidence
- Category
- Not specified
- Journal tier
- Top Field-Leading
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper (Tsujii–Zhang, 2020) proves the exact equivalence between topological mixing and exponential mixing for C∞ transitive Anosov flows on compact 3-manifolds (Theorem 1.1) and explains why (2) ⇒ (1) is immediate. The candidate solution’s (ii) ⇒ (i) argument via positivity of correlations for bump functions is standard and correct; for (i) ⇒ (ii) it cites the paper’s main theorem and sketches Dolgopyat-type transfer-operator ideas. A minor discrepancy is that the model mentions anisotropic spaces, whereas the paper emphasizes an approach via Markov partitions and complex RPF operators. Substantively, both reach the same conclusion, but by citation rather than reproducing the paper’s proof.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} top field-leading
\textbf{Justification:}
This paper establishes a landmark equivalence for C∞ transitive Anosov flows in dimension three, showing that topological mixing is precisely equivalent to exponential mixing for all Hölder equilibrium states and test functions. The approach synthesizes coding via Markov partitions with a refined Dolgopyat framework and a careful geometric analysis of uniform non-integrability, yielding robust L2 bounds for twisted transfer operators and, hence, a spectral decay. The arguments are intricate but coherent, with clear modular reductions. Minor clarifications on constant dependencies and a brief comparison with anisotropic-space methods would further enhance readability.