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),f) be the natural extension of (X, f), where K(X) is the family of all nonempty compact subsets of X. The above properties of f are further studied. It is proved that syndetical transitivity, total transitivity, and weak mixing of f are equivalent, and if f is strongly transitive, then f is mixing.

中文翻译：

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关于紧度量空间上连续映射的传递性和混合的进一步讨论

令 (X, d) 是一个紧致度量空间，而 f : X → X 是一个连续映射。这项工作研究了各种形式的传递性之间的关系，例如合合传递性、完全传递性和强传递性。得到了 f 完全（分别，强和合）传递，弱混合，混合和最小的一些必要和充分条件，或充分条件。然后，让(K(X),f) 是(X, f) 的自然扩展，其中K(X) 是X 的所有非空紧子集的族。进一步研究了f 的上述性质。证明了f的合称传递性、全传递性和弱混合性是等价的，如果f是强传递性，则f是混合性。