A characterization of nonautonomous attractors via Stone-Čech compactification

Abstract

The present paper deals with the notions of past attractors and repellers for nonautonomous dynamical systems. This uses the topological method of extending functions in order to describe the nonautonomous attractors by means of the prolongational limit sets in the extended phase space. Essentially, for a given nonautonomous dynamical system $(\theta ,\varphi ) $ with base set $P=\mathbb{T}$, where $\mathbb{T}$ is the time $\mathbb{Z}$ or $\mathbb{R}$, and with base flow $\theta $ as the addition, the limit sets $\omega ^{-}( 0) $ and $\omega^{+}( 0) $ in the Stone-Čech compactification $\beta \mathbb{T}$ determine respectively the past and the future of the conduction system.