Abstract
We consider products of an independent and identically distributed sequence in a set
$\{f_1,\ldots ,f_m\}$
of orientation-preserving diffeomorphisms of the circle. We can naturally associate a Lyapunov exponent
$\lambda $
. Under few assumptions, it is known that
$\lambda \leq 0$
and that the equality holds if and only if
$f_1,\ldots ,f_m$
are simultaneously conjugated to rotations. In this paper, we state a quantitative version of this fact in the case where
$f_1,\ldots ,f_m$
are
$C^k$
perturbations of rotations with rotation numbers
$\rho (f_1),\ldots ,\rho (f_m)$
satisfying a simultaneous diophantine condition in the sense of Moser [On commuting circle mappings and simultaneous diophantine approximations. Math. Z.205(1) (1990), 105–121]: we give a precise estimate of
$\lambda $
(Taylor expansion) and we prove that there exist a diffeomorphism g and rotations
$r_i$
such that
$\mbox {dist}(gf_ig^{-1},r_i)\ll |\lambda |^{{1}/{2}}$
for
$i=1,\ldots , m$
. We also state analogous results for random products of
$2\times 2$
matrices, without any diophantine condition.
Noncommuting random products