Abstract
This article studies past attractors and future repellers of topological cocycles conducted by a group G satisfying Ore's condition. The limit behaviour depends on the Green L -pre-order on a right reversible semigroup S of G. An existence and uniqueness theorem for past attractors is proved. A topological method of extending functions is used in order to describe the past attractors by means of skew-product prolongations in the extended space. Illustrative examples of past attractor and future repeller are provided.
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