On the ergodicity of anti-symmetric skew products with singularities and its applications

We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel-Cantelli-type arguments from Fayad and Lemańczyk (2006). The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and antisymmetric cocycles. Moreover, its most significant advantage is its ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles.

💡 Research Summary

The paper develops a new method for proving ergodicity of skew‑product extensions of interval exchange transformations (IETs) when the cocycle possesses singularities that are not limited to the classical logarithmic type. The authors introduce two auxiliary increasing C¹ functions, θ and τ (with τ(s)=s²·θ(1/s)), and define a function space Υ_θ consisting of functions that are C¹ on the interior of each exchanged interval but may have singular behavior at the interval endpoints governed by the growth of τ. A key quantitative condition, z_θ(f)>0, ensures that at least one endpoint exhibits a non‑trivial singularity.

The central result (Theorem 1.1) states that for almost every symmetric IET (i.e., the permutation satisfies π₀(α)=π₁(α)¯¹) and for any cocycle f∈Υ_θ satisfying three properties—(i) z_θ(f)>0, (ii) anti‑symmetry f∘T⁻¹∘I=−f (where I is the central reflection), and (iii) piecewise monotonicity on each interval—the skew product T_f(x,r)=(T x, r+f(x)) is ergodic. The proof adapts the Borel‑Cantelli arguments of Fayad and Lemańczyk (2006) to the IET setting, using Rauzy‑Veech induction to control return times and to show that large jumps caused by the singularities occur rarely enough to satisfy a Borel‑Cantelli criterion. The anti‑symmetry condition forces the average of the cocycle to vanish, which is crucial for constructing independent “large‑jump” events.

The authors then apply this abstract ergodicity criterion to locally Hamiltonian flows ψ_t on compact orientable surfaces of genus g≥1. By taking a transversal interval I, the first‑return map of ψ_t to I is a symmetric IET, and the first‑return time τ_I gives rise to a cocycle φ_f(x)=∫₀^{τ_I(x)} f(ψ_s x) ds. Previous works (Forni, Bufetov, Marmi‑Moussa‑Yoccoz, etc.) established ergodicity and equidistribution of the error term in Birkhoff integrals only when φ_f has symmetric logarithmic singularities, a situation that requires all saddles to be perfect and the absence of saddle loops. The new theorem allows φ_f to have more general singularities of the form s·θ(1/s), thereby covering cases where some saddles are imperfect (degenerate) or where saddle loops break the symmetry of the logarithmic singularities.

The paper proves two concrete applications. First, Theorem 9.1 shows that for locally Hamiltonian flows without saddle loops (i.e., minimal on the whole surface), the error term in the deviation spectrum is equidistributed even when the cocycle’s singularities are not purely logarithmic. Second, Theorem 2.4 extends this equidistribution to hyper‑elliptic flows with perfect saddles but with logarithmic singularities that are not symmetric, demonstrating that the symmetry condition is not essential for ergodicity. In Appendix A the authors construct a family of flows with a new type of degenerate, imperfect saddles and prove that the associated anti‑symmetric skew products are ergodic, providing the first examples of such behavior beyond rotation extensions.

Technically, the paper’s contributions are threefold: (1) a generalized singularity control framework based on θ and τ, (2) the exploitation of anti‑symmetry to obtain zero‑mean cocycles, and (3) a refined use of Rauzy‑Veech dynamics to establish the required probabilistic independence for Borel‑Cantelli arguments. These tools together allow the authors to remove the restrictive logarithmic‑singularity assumption and to treat flows with a broader class of saddle configurations.

In summary, the work significantly broadens the class of skew‑product systems over IETs for which ergodicity can be established, and it translates this abstract result into concrete statistical statements about Birkhoff sums for locally Hamiltonian flows on surfaces, even in the presence of non‑perfect saddles or broken symmetry. This advances our understanding of how subtle non‑symmetric singularities influence long‑term statistical properties in low‑dimensional dynamical systems.