Singularity of the spectrum of typical minimal smooth area-preserving flows in any genus

TL;DR

The paper proves that for genus $g\geq 2$, a typical smooth area-preserving flow on a closed surface with only simple saddles has purely singular spectrum, and that almost every pair of such flows is spectrally disjoint. It accomplishes this by representing locally Hamiltonian flows as special flows over IETs with symmetric logarithmic roofs, and then proving a singularity criterion based on global rigidity plus exponential-tailed distributions of Birkhoff sums, together with a new disjointness mechanism via resonant rigid times and shearing. The proof hinges on a deep analysis of Rauzy-Veech induction, including balanced and accelerated inductions, refined control of trimmed Birkhoff sums of derivatives, and a probabilistic, Markovian treatment of cocycles to produce the necessary rigidity-events and resonant times. The results extend the understanding of spectral and disjointness phenomena in parabolic flows beyond genus two, and establish a robust framework for singular spectral behavior in higher genus locally Hamiltonian dynamics with simple saddles. This work has implications for the spectral theory of area-preserving flows, the structure of interval exchange transformations, and the broader study of rigidity, mixing, and joining properties in zero-entropy dynamical systems.

Abstract

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that almost every such locally Hamiltonian flow with only simple saddles has singular spectrum. Furthermore, we prove that almost every pair of such flows is spectrally disjoint. More in general, singularity of the spectrum and pairwise disjointness holds for special flows over a full measure set of interval exchange transformations under a roof with symmetric logarithmic singularities. The spectral result is proved using a criterion for singularity based on tightness of Birkhoff sums with exponential tails decay and the cancellations proved by the last author to prove absence of mixing in this class of flows, by showing that the latter can be combined with rigidity. Disjointness of pairs then follows by producing mixing times (for the second flow), using a new mechanism for shearing based on resonant rigidity times.

Figures (3)

• Figure 1: Type of fixed points for area-preserving flows.
• Figure 2: Trajectories of a locally Hamiltonian flow with four simple saddles on a surface of genus three.
• Figure 3: A special flow over a $7$-IET under a roof $f \in \mathcal{S}\textrm{ym}\mathcal{L}\textrm{og} \left( T \right)$ which is a special representation of a locally Hamiltonian flow with two saddles of multiplicity $2$ on a surface of genus $3$.

Theorems & Definitions (80)

• Theorem 1: Singular spectrum of typical higher genus flows with simple saddles
• Theorem 2
• Theorem 3
• Remark 2.1
• Proposition 2.1
• Proposition 2.2
• proof : Proof of Theorem \ref{['thm:singsp']} and Theorem \ref{['thm:mixing_disj']}
• proof : Proof of Theorem \ref{['thm:main_dj']}.
• Definition 4.1: Rigidity
• Theorem 4: Singularity Criterion via rigidity and exponential tails, see Theorem 3.1 in Ch-Fr-Ka-Ul