A new method to prove the nonuniform dichotomy spectrum theorem in ℝⁿ

Abstract

This paper presents a new method to prove the nonuniform dichotomy spectrum theorem. Chu et al. [Bull. Sci. Math. 139 (2015), pp. 538–557] and Zhang [J. Funct. Anal. 267 (2014), pp. 1889–1916] generalized the dichotomy spectrum in Siegmund [J. Dynam. Differential Equations 14 (2002), pp. 243–258] to the nonuniform dichotomy spectrum and the authors in these works employed linear integral manifolds (stable and unstable) to establish the spectral theorem. They then used the spectrum theorem to study reducibility. We prove the nonuniform dichotomy spectrum by way of contradiction. In particular, we employ the nonuniform kinematically similarity (nonuniform reducibility) to reduce the shift system into two blocks and then we get a contradiction based on a technique in mathematical analysis. The method in the proof is completely different from previous works.