Communications in Mathematical Physics

, Volume 332, Issue 3, pp 1113–1166 | Cite as

Continuity of the Lyapunov Exponents for Quasiperiodic Cocycles

  • Pedro Duarte
  • Silvius KleinEmail author


Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Hölder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Diophantine translations on higher dimensional tori, albeit with a loss in the modulus of continuity of the Lyapunov spectrum blocks.


Manifold Lyapunov Exponent Subharmonic Function Lyapunov Spectrum Base Dynamic 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Departamento de Matemática and CMAF, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal
  2. 2.CMAF, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal
  3. 3.IMARBucharestRomania

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