A generalized Poincaré stability criterion

Abstract:

Let $\Phi _t^\# \eta = {\Phi ^{ - t}} \circ \eta \circ {\phi ^t}$ define a semigroup on the Banach space $\Gamma (M,E)$ of continuous sections of $E$ over $M$. It is known that $({\Phi ^t},{\phi ^t})$ is hyperbolic iff $\Phi _\# ^t$ has spectrum off the unit circle for $t \ne 0$. We prove that a third equivalent condition is that the (unbounded!) infinitesimal generator $L$ of $\{ \Phi _t^\# \}$ have its spectrum disjoint from the imaginary axis. In two dimensions this property coincides with the Poincaré stability criterion for a periodic orbit of a planar dynamical system.