Abstract
Let (X, d) be a compact metric space and f : X → X be a continuous map. This work investigates the relationships among various forms of transitivity, such as syndetically transitive, totally transitive, and strongly transitive. Some necessary and sufficient conditions, or sufficient conditions for f to be totally (respectively, strongly and syndetically) transitive, weakly mixing, mixing, and minimal, are obtained. Then, let (K(X),f̄) be the natural extension of (X, f), where K(X) is the family of all nonempty compact subsets of X. The above properties of f̄ are further studied. It is proved that syndetical transitivity, total transitivity, and weak mixing of f̄ are equivalent, and if f̄ is strongly transitive, then f̄ is mixing.
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