Stability of group representations and Haar spectrum

Abstract:

If $U$ and $V$ are commuting unitary representations of locally compact abelian groups $S$ and $T$, new representations of $S$ (perturbations of $U$) can be obtained from composition with images of $U$ in $V$. If most of these representations are equivalent to $U,U$ is said to be $V$ stable. We investigate conditions which, together with stability, ensure that $U$ has (uniform) Haar spectrum. The principal applications are to dynamical systems which possess auxiliary groups with respect to which motion is stable.