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NONUNIFORM EXPONENTIAL BEHAVIOR VIA EVOLUTION SEMIGROUPS
Author(s) -
Barreira Luis,
Popescu Liviu Horia,
Valls Claudia
Publication year - 2020
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12001
Subject(s) - mathematics , exponential function , exponential formula , exponential growth , pure mathematics , double exponential function , mathematical analysis
We provide a unified framework to study the nonuniform exponential behavior of a dynamics using the theory of evolution semigroups. We emphasize that nonuniform hyperbolicity is ubiquitous in the context of ergodic theory: for an autonomous differential equation whose flow preserves a finite measure, the variational equations of almost all trajectories with nonzero Lyapunov exponents have a nonuniform exponential behavior. We consider the general case of a nonautonomous dynamics on a Banach space. In particular, we show that to any nonuniformly exponentially bounded evolution family, one can associate a C 0 semigroup (thus an autonomous dynamics) on some appropriate Banach space. This semigroup is then used to characterize completely the nonuniform hyperbolicity of the evolution family. In addition, we establish a corresponding spectral mapping theorem for an evolution semigroup, which allows us to obtain another characterization of the nonuniform hyperbolicity of an evolution family in terms of the spectrum of the evolution semigroup.