Linearized stability in the context of an example by Rodrigues and Solà-Morales

Abstract

In a recent paper, Rodrigues and Solà-Morales construct an example of a continuously Fréchet differentiable discrete dynamical system in a separable Hilbert space for which the origin is an exponentially asymptotically stable fixed point, although its derivative at 0 has spectral radius greater than one. For maps on general Banach spaces we demonstrate that the slightly stronger, but also widely used concept of exponential stability allows a complete characterization in terms of the spectral radius. Moreover, under a spectral gap condition valid for compact and finite-dimensional linearizations these two stability notions are shown to be equivalent.