For a continuous self-map f on a compact metric space X, we provide two simple examples: the first confirms that shadowing of (X, f) is not inherited by \(({\mathcal {M}}(X),{\hat{f}})\) in general, and the other satisfies that both (X, f) and \(({\mathcal {M}}(X),{\hat{f}})\) have no Li–Yorke pair, where \({\mathcal {M}}(X)\) be the space of all Borel probability measures on X. Then we prove that chain transitivity of (X, f) implies chain mixing of \(({\mathcal {M}}(X),{\hat{f}})\), and provide an example to deny the converse. For a non-autonomous system \((X,f_{0,\infty })\), we prove that weak mixing of \(({\mathcal {M}}(X),\hat{f}_{0,\infty })\) implies that of \((X,f_{0,\infty })\), and give an example to deny the converse, where \(f_{0,\infty }=\{f_n\}_{n=0}^\infty \) is a sequence of continuous self-maps on X. We also prove that if \(f_n\) is surjective for all \(n\ge 0\), then chain mixing of \(({\mathcal {M}}(X),{\hat{f}}_{0,\infty })\) always holds true, and shadowing of \(({\mathcal {M}}(X),{\hat{f}}_{0,\infty })\) implies mixing of \((X, f_{0,\infty })\). If \(X=I\) is an interval, we obtain a sharp condition such that transitivity is equivalent between (I, f) and \(({\mathcal {M}}(I),{\hat{f}})\). Although \(({\mathcal {M}}(I),{\hat{f}})\) has infinite topological entropy for any transitive system (I, f), we give an example such that \((I,f_{0,\infty })\) is transitive but \(({\mathcal {M}}(I),{\hat{f}}_{0,\infty })\) has zero topological entropy.

对于紧凑度量空间X上的连续自映射f，我们提供两个简单的示例：第一个示例确认（X，  f）的阴影不被\（（{{mathcal {M}}（X），{ \帽子{F}}）\）在一般情况下，与其他满足这两个（X，  ˚F）和\（（{\ mathcal {M}}（X）{\帽子{F}}）\）没有锂约克对，其中\（{\ mathcal {M}}（X）\）是关于所有的Borel概率测度空间X。然后我们证明（X，  f）的链传递性意味着\（（{{数学{M}}（X），{\ hat {f}}）\）\）的链混合，并提供一个示例来说明相反的情况。对于非自治系统\（（X，f_ {0，\ infty}）\），我们证明了\（（{\ mathcal {M}}（X），\ hat {f} _ {0 ，\ infty}} \）表示\（（X，f_ {0 ，\ infty}）\）的含义，并举例说明拒绝进行相反的处理，其中\（f_ {0，\ infty} = \ {f_n \ } _ {n = 0} ^ \ infty \）是X上一系列连续的自映射。我们还证明，如果\（f_n \）对于所有\（n \ ge 0 \）都是射影，则\（（{{mathcal {M}}（X），{\ hat {f}} __ { 0，\ infty}} \）始终为真，\（（{\ mathcal {M}}（X），{\ hat {f}} _ {0，\ infty}）\\的阴影表示\ （（X，f_ {0，\ infty}）\）。如果\（X = I \）是一个间隔，我们得到一个尖锐的条件，使得（I，  f）和\（（{{\ mathcal {M}}（I），{\ hat {f}} ）\）。尽管\（（{{mathcal {M}}（I），{\ hat {f}}）\）对于任何传递系统（I，  f）都有无限大的拓扑熵，但我们举一个例子，使\（（I， f_ {0，\ infty}} \）是可传递的，但是\（（{{mathcal {M}}（I），{\ hat {f}} _ {0，\ infty}} \\}的拓扑熵为零。