[論文レビュー] 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被引用数 0

ひとこと要約

この論文は、Abelian カバー上の体積保存型 Anosov 流の等尺延長の相関関数に対する逆時的漸近展開を証明し、明示的な主項と残差推定を得る。

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.

研究の動機と目的

• 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.

提案手法

• 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.

実験結果

リサーチクエスチョン

• 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?

主な発見

• 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.