Zero Lyapunov exponents of the Hodge bundle

Giovanni Forni, Carlos Matheus, Anton Zorich

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)

Abstract

By the results of G. Forni and of R. Treviño, the Lyapunov spectrum of the Hodge bundle over the Teichmüller geodesic flow on the strata of Abelian and of quadratic differentials does not contain zeroes even though for certain invariant submanifolds zero exponents are present in the Lyapunov spectrum. In all previously known examples, the zero exponents correspond to those PSL(2, ℝ)-invariant subbundles of the real Hodge bundle for which the monodromy of the Gauss-Manin connection acts by isometries of the Hodge metric. We present an example of an arithmetic Teichmüller curve, for which the real Hodge bundle does not contain any PSL(2, ℝ)-invariant, subbundles, and nevertheless its spectrum of Lyapunov exponents contains zeroes. We describe the mechanism of this phenomenon; it covers the previously known situation as a particular case. Conjecturally, this is the only way zero exponents can appear in the Lyapunov spectrum of the Hodge bundle for any PSL(2, ℝ)-invariant probability measure.

Original languageEnglish
Pages (from-to)489-535
Number of pages47
JournalCommentarii Mathematici Helvetici
Volume89
Issue number2
DOIs
Publication statusPublished - 2014
Externally publishedYes

Keywords

  • Groups of pseudo-unitary complex matrices
  • Hodge bundle
  • Kontsevich-Zorich cocycle
  • Lyapunov exponents
  • Moduli spaces of Abelian and quadratic differentials
  • Teichmüller flow

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