Abstract
In this paper, we introduce the notions of weakly mixing and totally transitivity for a free semigroup action. Let [Formula: see text] be a free semigroup acting on a compact metric space generated by continuous open self-maps. Assuming shadowing for [Formula: see text] we relate the average shadowing property of [Formula: see text] to totally transitivity and its variants. Also, we study some properties such as mixing, shadowing and average shadowing properties, transitivity, chain transitivity, chain mixing and specification property for a free semigroup action.
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