Abstract
The aim of this work is to study the -bi-shadowing property of the metric -space. We generalized the results to the metric -space and studied the chaotic properties by introducing a new definition of chaos which we call --chaotic in a neighborhood of a set and comparing it with the definition of Li-York of chaos in -space. We will study the above definitions with -homoclinic orbit and -chain components.
 The main results that we obtained in this paper, for some conditions, is -the homoclinic orbit of, and is both --bi-shadowing and --periodic bi-shadowing on (when being an unordered set), then any action which satisfies some conditions is --chaotic on a neighborhood of. Second, for some conditions, if action is --expansive and both --bi-shadowing and --periodic bi-shadowing with respect to an action and is a -the homoclinic orbit of contained, then every action satisfying some conditions is --chaotic on a neighborhood of. Third, for some conditions, if be a -chain component of an action, and is both --bi-shadowing and --periodic bi-shadowing on a -chain recurrent set -, Then every action which satisfies some conditions is --chaotic in a neighborhood.
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