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PROC

Please refer to Full Text.

keywords:

JMD

If $T=\{T_t\}_{t\in\mathbb R}$ is an aperiodic
measure-preserving jointly
continuous flow on a compact metric space $\Omega$ endowed with a Borel
probability measure $m$, and $G$ is a compact Lie group with Lie
algebra $L$, then to each continuous map $A: \Omega \to L$
associate the solution $\Omega\times\mathbb R$ ∋ $(\omega,t)\mapsto X^A(\omega,t)\in G$
of the family of time-dependent initial-value problems $X'(t) =
A(T_t\omega)X(t)$, $X(0) =$ identity, $X(t) \in G$ for
$\omega\in \Omega$.
The corresponding skew-product flow $T^A=\{T_t^A\}_{t\in\mathbb R}$ on
$G\times\Omega$ is then defined by letting $T^A_t(g,\omega ) = (X^A(\omega
,t)g,T_t\omega)$ for $(g,\omega)\in G\times\Omega$, $t\in\mathbb R$. The flow
$T^A$ is measure-preserving on $(G\times \Omega,\nu_{_G}\otimes m)$ (where
$\nu_{_G}$ is normalized Haar measure on $G$) and jointly continuous.
For a given
closed convex subset $S$ of $L$, we study the set $C_{erg}(\Omega ,S)$
of all continuous maps $A: \Omega\to S$ for which the flow $T^A$ is
ergodic. We develop a new technique to determine a necessary and sufficient
condition for the set $C_{erg}(\Omega ,S)$ to be residual.
Since the dimension of $S$ can be much
smaller than that of $L$, our result proves that ergodicity is typical
even within very "thin'' classes of cocycles. This covers a number of
differential equations arising in mathematical physics, and in
particular applies to the widely studied example of the Rabi
oscillator.

CPAA

We show that a generic $SL(2,R)$ valued cocycle in the class of
$C^r$, ($0 < r < 1$) cocycles based on a rotation flow on the $d$-torus, is either uniformly hyperbolic or has zero Lyapunov exponents provided that the components of winding vector $\bar \gamma = (\gamma^1,\cdot \cdot \cdot,\gamma^d)$ of the rotation flow are rationally independent and
satisfy the following super Liouvillian condition :

$ |\gamma^i - \frac{p^i_n}{q_n}| \leq Ce^{-q^{1+\delta}_n}, 1\leq i\leq d, n\in N,$

where $C > 0$ and $\delta > 0$ are some constants and $p^i_n, q_n$ are some sequences of integers with $q_n\to \infty$.

## Year of publication

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