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Hölder regularity for the spectrum of translation flows

Alexander I. Bufetov, Boris Solomyak

TL;DR

The paper addresses Hölder regularity of spectral measures for translation flows arising from Abelian differentials on flat surfaces of genus $g\ge 2$, extending prior genus-2 results to all genera. It combines Forni’s vector Erdős–Kahane approach with symbolic dynamics of random $S$-adic (Markov) systems, proving Hölder bounds for spectral measures for almost every abelian differential in a stratum. The method hinges on sharp bounds for twisted Birkhoff integrals, a quantitative Veech criterion, and an Erdős–Kahane-type dimension estimate for the exceptional set in the strong unstable subspace of the Kontsevich–Zorich cocycle. This yields quantitative weak mixing information and Hölder regularity for flows on random Markov compacta, with implications for translation flows on surfaces of arbitrary genus and connections to moduli dynamics. The work thereby unifies symbolic and geometric techniques to obtain robust regularity results with potential extensions to random tilings and higher-dimensional flat dynamics.

Abstract

The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus $g\ge 2$. These flows are weakly mixing by the Avila-Forni theorem. In genus 2, the Hölder property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained Hölder estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni's idea with the symbolic approach of [10] and prove Hölder regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera.

Hölder regularity for the spectrum of translation flows

TL;DR

The paper addresses Hölder regularity of spectral measures for translation flows arising from Abelian differentials on flat surfaces of genus , extending prior genus-2 results to all genera. It combines Forni’s vector Erdős–Kahane approach with symbolic dynamics of random -adic (Markov) systems, proving Hölder bounds for spectral measures for almost every abelian differential in a stratum. The method hinges on sharp bounds for twisted Birkhoff integrals, a quantitative Veech criterion, and an Erdős–Kahane-type dimension estimate for the exceptional set in the strong unstable subspace of the Kontsevich–Zorich cocycle. This yields quantitative weak mixing information and Hölder regularity for flows on random Markov compacta, with implications for translation flows on surfaces of arbitrary genus and connections to moduli dynamics. The work thereby unifies symbolic and geometric techniques to obtain robust regularity results with potential extensions to random tilings and higher-dimensional flat dynamics.

Abstract

The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus . These flows are weakly mixing by the Avila-Forni theorem. In genus 2, the Hölder property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained Hölder estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni's idea with the symbolic approach of [10] and prove Hölder regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera.

Paper Structure

This paper contains 27 sections, 18 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Formulation of the main result
  3. Quantitative weak mixing
  4. Organization of the paper, comparison with BuSo18a and BuSo19
  5. Further directions
  6. Preliminaries
  7. Markov compacta
  8. Vershik's orderings and Vershik automorphisms
  9. The space of Markov compacta and the renormalization cocycle
  10. Substitutions and $S$-adic systems
  11. Suspension flows and cylindrical functions
  12. Shift for $S$-adic systems
  13. Telescoping an $S$-adic sequence BSTY
  14. Weakly Lipschitz functions.
  15. Interval exchange transformations and suspensions over them
  16. ...and 12 more sections

Key Result

Theorem 1.1

There exists $\gamma>0$ such that for $\mu_{\mathcal{H}}$-almost every abelian differential $(M, \boldsymbol \omega)\in {\mathcal{H}}$ the following holds. For any $B>1$ there exist constants $C=C(\boldsymbol \omega,B)$ and $r_0=r_0(\boldsymbol \omega,B)$ such that for any Lipschitz function $f$ on

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Theorem 2.2: BSTY
  • Theorem 2.3: BSTY
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 23 more