| | |
Summary: EXTREMAL LYAPUNOV EXPONENTS OF SMOOTH
COCYCLES
ARTUR AVILA 1,2 AND MARCELO VIANA 2
Abstract. A smooth cocycle is a skew-product map that acts by diffeomor-
phisms on the fibers. The smallest and largest Lyapunov exponents measure
the minimum and maximum exponential rates of variation of the norm of the
derivative along the fibers. We discuss conditions under which these numbers
may vanish. The approach is based on a non-linear extension of a classical
result of Ledrappier that we state and prove in here. The main applications
in the present paper are for area preserving fiber bunched cocycles. Analyzing
Lyapunov exponents as functions of the cocycle, we find that the points of dis-
continuity are rather rigid, even more so in the accessible case: in particular,
the Oseledets decomposition must be continuous. In addition, we prove that
for an open dense subset the Lyapunov exponents are different from zero and
vary continuously with the cocycle.
Contents
1. Introduction 2
2. Non-linear invariance criterion 8
3. Cocycles with invariant holonomies 18
4. Domination and fiber bunching 24
|
