[Paper Review] H\\"older regularity for the spectrum of translation flows

Alexander I. Bufetov, Boris Solomyak|arXiv (Cornell University)|Aug 25, 2019

Mathematical Dynamics and Fractals19 references4 citations

TL;DR

This paper establishes Hölder regularity for the spectral measures of translation flows on flat surfaces of genus $ g \geq 2 $, combining Forni's vector-valued Erdős-Kahane argument with symbolic dynamics on random Markov compacta. The key result is that for almost every abelian differential under the Masur-Veech measure, spectral measures of Lipschitz functions satisfy $ \sigma_f([\lambda - r, \lambda + r]) \leq C \|f\|_L r^\gamma $ for some $ \gamma > 0 $, confirming a sharp Hölder estimate that quantifies weak mixing.

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 H\\"older property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained H\\"older 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 H\\"older regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera.

Motivation & Objective

• To establish Hölder regularity for spectral measures of translation flows on compact flat surfaces of genus $ g \geq 2 $, extending prior results from genus 2 to general genus.
• To unify Forni's analytic approach with the symbolic framework of Bufetov and others, simplifying the proof via vector-form Erdős-Kahane estimates.
• To demonstrate that the Hölder exponent $ \gamma > 0 $ holds for almost every abelian differential with respect to the Masur-Veech measure.
• To extend the result beyond the Masur-Veech measure to a broader class of invariant measures under the Teichmüller flow, provided conditional measures on cohomology fibers have sufficient Hausdorff dimension.

Proposed method

• Uses the symbolic representation of interval exchange transformations via Rauzy-Veech induction and admissible words in the Rauzy graph to model translation flows on random Markov compacta.
• Applies a vector-valued version of the Erdős-Kahane argument to control twisted Birkhoff integrals, replacing scalar estimates used in earlier work.
• Employs return words and substitution dynamics to construct good return times and analyze the growth of cylinder sets in the symbolic system.
• Leverages the simplicity of admissible words and positivity of substitution matrices to ensure full rank generation of integer lattices in cohomology.
• Relies on the Teichmüller flow's exponential mixing and compactness estimates to control return times and spectral decay.
• Establishes the Hölder exponent $ \gamma > 0 $ via uniform bounds on spectral measures over small intervals, using the structure of the Rauzy-Veech renormalization and cohomological coordinates.

Experimental results

Research questions

• RQ1 Does the spectral measure of a Lipschitz function under a generic translation flow on a genus $ g \geq 2 $ surface satisfy a Hölder-type regularity condition?
• RQ2 Can the Hölder regularity of spectral measures be established using a vector-form of the Erdős-Kahane argument in a symbolic dynamical framework?
• RQ3 To what extent does the result extend beyond the Masur-Veech measure to other invariant measures under the Teichmüller flow?
• RQ4 What is the minimal regularity condition on conditional measures in cohomology fibers that ensures the Hölder property for spectral measures?

Key findings

• For $ \mu_{\mathcal{H}} $-almost every abelian differential $ (M, \boldsymbol{\omega}) \in \mathcal{H} $, the spectral measure $ \sigma_f $ of any Lipschitz function $ f $ satisfies $ \sigma_f([\lambda - r, \lambda + r]) \leq C \|f\|_L r^\gamma $ for all $ r \in (0, r_0) $, with $ \gamma > 0 $, $ C = C(\boldsymbol{\omega}, B) $, and $ \lambda \in [B^{-1}, B] $.
• The Hölder exponent $ \gamma > 0 $ is uniform in $ \lambda $ over compact sets $ [B^{-1}, B] $, and the constant $ C $ depends on the differential and the bound $ B $.
• The result holds for a broader class of invariant measures $ \mu $ under the Teichmüller flow, provided the conditional measure on cohomology fibers has Hausdorff dimension at least $ 2g - \kappa + \delta $ for some $ \delta > 0 $, where $ \kappa $ is the number of positive Lyapunov exponents.
• The proof simplifies the original approach in [12, 14] by replacing scalar estimates with vector-valued Erdős-Kahane arguments, yielding a more robust and general framework.
• The existence of a simple admissible word $ \mathbf{q} $ with a strictly positive substitution matrix and generating return words whose population vectors span $ \mathbb{Z}^m $ ensures the necessary algebraic structure for the argument.
• The result confirms that the spectral measures of translation flows are not only singular but also exhibit a sharp Hölder regularity, quantifying the degree of weak mixing in these systems.