New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows

TL;DR

This paper investigates new deviation phenomena in Birkhoff integrals for locally Hamiltonian flows on surfaces, revealing additional spectral terms caused by non-vanishing derivatives at fixed points, extending previous results to more complex singularities.

Contribution

It extends the deviation spectrum analysis of Birkhoff integrals to include effects of non-vanishing derivatives at fixed points and introduces new methods for handling polynomial singularities.

Findings

Existence of extra deviation spectrum terms due to non-vanishing derivatives.

Extension of deviation spectrum results to flows with degenerate saddles.

Development of new techniques for functions with polynomial singularities.

Abstract

We consider smooth locally Hamiltonian flows on compact surfaces of genus $g\geq 2$ to prove their deviation of Birkhoff integrals for smooth observables. Our work generalizes results of Forni and Bufetov which prove the existence of deviation spectrum of Birkhoff integrals for observables whose jets vanish at sufficiently high order around fixed points of the flow. They showed that ergodic integrals can display a power spectrum of behaviours with exactly $g$ positive exponents related to the positive Lyapunov exponents of the cocycle so-called Kontsevich-Zorich, a renormalization cocycle over the Teichm\"uller flow on a stratum of the moduli space of translation surfaces. Our paper extends the study of the spectrum of deviations of ergodic integrals beyond the case of observables whose jets vanish at sufficiently high order around fixed points. We prove the existence of some extra terms in deviation spectrum related to non-vanishing of the derivatives of observables at fixed points. The proof of this new phenomenon is based on tools developed in the recent work of the first author and Ulcigrai for locally Hamiltonian flows having only (simple) non-degenerate saddles. In full generality, due to the occurrence of (multiple) degenerate saddles, we introduce new methods of handling functions with polynomial singularities over a full measure set of IETs.