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Conditions for equality between Lyapunov and Morse decompositions

Published online by Cambridge University Press:  08 January 2015

LUCIANA A. ALVES  and
LUIZ A. B. SAN MARTIN
LUCIANA A. ALVES
Affiliation:
Faculdade de Matemática – Universidade Federal de Uberlândia, Campus Santa Mônica, Av. João Naves de Ávila, 2121, 38408-100 Uberlândia – MG, Brasil email lualves@famat.ufu.br
LUIZ A. B. SAN MARTIN
Affiliation:
Imecc – Unicamp, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz, 13083-859 Campinas – SP, Brasil email smartin@ime.unicamp.br
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Abstract

Let $Q\rightarrow X$ be a continuous principal bundle whose group $G$ is reductive. A flow ${\it\phi}$ of automorphisms of $Q$ endowed with an ergodic probability measure on the compact base space $X$ induces two decompositions of the flag bundles associated to $Q$: a continuous one given by the finest Morse decomposition and a measurable one furnished by the multiplicative ergodic theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under perturbations leaving unchanged the flow on the base space.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 4 , June 2016 , pp. 1007 - 1036
Copyright
© Cambridge University Press, 2015 

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