April  2007, 1(2): 205-253. doi: 10.3934/jmd.2007.1.205

Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form

1. 

Department of Mathematics, Rutgers University, Camden NJ 08102, United States

2. 

Department of Mathematics, Rutgers University, Piscataway, NJ 08854, United States

Received  May 2006 Published  January 2007

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.
Citation: Mahesh G. Nerurkar, Héctor J. Sussmann. Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form. Journal of Modern Dynamics, 2007, 1 (2) : 205-253. doi: 10.3934/jmd.2007.1.205
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