Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmuller flow

• 成果类型: Article
• 署名作者: Avila, Artur; Gouezel, Sebastien
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2013.178.2.1
• 发表日期: 2013
• 页码: 385-442
• 关键词: connected components; transformations

摘要

We consider the SL(2; R) action on moduli spaces of quadratic differentials. If mu is an SL(2; R)-invariant probability measure, crucial information about the associated representation on L-2(mu) (and, in particular, fine asymptotics for decay of correlations of the diagonal action, the Teichmuller flow) is encoded in the part of the spectrum of the corresponding foliated hyperbolic Laplacian that lies in (0, 1/4) (which controls the contribution of the complementary series). Here we prove that the essential spectrum of an invariant algebraic measure is contained in [1/4, infinity); i.e., for every infinity > 0, there are only finitely many eigenvalues (counted with multiplicity) in (0, 1/4-delta). In particular, all algebraic invariant measures have a spectral gap.