[论文解读] Decay of correlations on Abelian covers of isometric extensions of volume-preserving Anosov flows

Mihajlo Cekić, Thibault Lefeuvre+1 more|arXiv (Cornell University)|Mar 6, 2026

Mathematical Dynamics and Fractals被引用 0

一句话总结

该论文证明了在阿贝尔覆盖上的等距扩展的体积保持型有同变流的相关函数的反时间渐近展开，给出明确的主项和剩余项估计。

ABSTRACT

We establish an asymptotic expansion in inverse powers of time of the correlation function of isometric extensions of volume-preserving Anosov flows on Abelian covers of closed manifolds.

研究动机与目标

• Motivate and study decay of correlations for Anosov flows on Abelian covers.
• Extend decay results to isometric G-extensions with compact Lie groups over Abelian covers.
• Obtain an explicit asymptotic expansion in inverse powers of time for correlation functions.
• Characterize how the leading term depends on the trivial representation and the dynamical connection.

提出的方法

• Use Floquet theory and Abelian covers to decompose functions into Fourier modes on the cover.
• Develop an isometric extension framework with a compact Lie group and dynamical connection.
• Employ semiclassical and Borel-Weil calculus on G-bundles to analyze equivariant operators.
• Derive a time-asymptotic expansion t^{d/2} correlation[ f ∘ φ_{-t}, g ] = κ ∫f ∫g + ∑_{j≥1} t^{-j} C_j(f,g) + R_N(t,f,g).
• Provide explicit bounds for C_j and R_N in terms of Grönwall-type norms B^{s,r}(M) and B^{s,·}(M).
• Specialize to frame flows to obtain Corollary 1.3 and connect to ergodicity conditions.

实验结果

研究问题

• RQ1 What is the asymptotic decay rate of correlations for Abelian covers of isometric extensions of volume-preserving Anosov flows?
• RQ2 How does the leading term and higher-order corrections depend on the trivial representation and the dynamical connection?
• RQ3 Can one obtain a full asymptotic expansion for correlations on G-extensions over Abelian covers using Borel-Weil calculus?
• RQ4 What are the precise remainder estimates and regularity requirements for the expansion to hold?
• RQ5 How do these results apply to specific flows such as frame flows on negatively curved manifolds?

主要发现

• There exists a=d-based decay of correlations expansion with leading term proportional to t^{-d/2} for Abelian covers when dα ≠ 0.
• The expansion includes explicit bilinear forms C_j(f,g) for j≥1 with nonzero C_j (j≥1).
• The remainder R_N(t,f,g) decays like ⟨t⟩^{-N} with norms in the B^{s,r} scale controlling the bounds.
• The leading constant κ is explicit and involves a (2π)^{d/2} factor and the inverse square root of a covariance determinant.
• Theorem 1.2 extends the result to isometric G-extensions with the transitivity group H = G and independence of the forms F_i.
• Corollary 1.3 specializes these results to frame flows under ergodicity conditions, yielding a similar expansion for FN.