AI 中文总结
研究集值映射及其逆极限的跟踪性、传递性等性质，证明了满射上半连续集值映射与其逆极限的跟踪性等价，且广义逆极限上移位映射的传递性等性质能推出集值映射相应性质，还给出了具有跟踪性的集值映射相关性质的等价关系。
AI 中文摘要
我们研究了集值映射及其广义逆极限的跟踪性、传递性、弱混合性、混合性、链传递性和链混合性等性质。关于跟踪性，证明了在紧致度量空间上的满射上半连续集值映射\(F\)具有跟踪性当且仅当其逆集值映射的广义逆极限\(\underleftarrow{\lim}\,\underleftarrow{F}\)上的移位映射具有跟踪性；反之，\(\underleftarrow{F}\)具有跟踪性当且仅当\(\underleftarrow{\lim}F\)上的移位映射具有跟踪性。进一步表明\(F\)和\(\underleftarrow{F}\)的跟踪性总是等价的，从而\(F\)、\(\underleftarrow{F}\)以及\(\underleftarrow{\lim}\,\underleftarrow{F}\)和\(\underleftarrow{\lim}F\)上的移位映射都同时具有跟踪性。特别地，\(F\)具有跟踪性当且仅当其直接诱导的广义逆极限\(\underleftarrow{\lim}F\)上的移位映射具有跟踪性。还证明了如果广义逆极限上的移位映射是传递的（分别对应弱混合、混合、链传递、链混合），那么集值映射也是传递的（分别对应弱混合、混合、链传递、链混合）。对于具有跟踪性的集值映射，全传递性、弱混合性、混合性、规范性质和链混合性是相互等价的。
英文摘要
We study the properties of shadowing, transitivity, weakly mixing, mixing, chain transitivity and chain mixing of a set-valued map and its generalized inverse limit. Concerning shadowing, we prove that for a surjective upper semi-continuous set-valued map $F$ on a compact metric space, $F$ has shadowing if and only if the shift map on the generalized inverse limit $\underleftarrow{\lim}\,\underleftarrow{F}$ of its inverse set-valued map has shadowing; dually, $\underleftarrow{F}$ has shadowing if and only if the shift map on $\underleftarrow{\lim}F$ has shadowing. We further show that the shadowing of $F$ and that of $\underleftarrow{F}$ are always equivalent; consequently $F$, $\underleftarrow{F}$ and the shift maps on $\underleftarrow{\lim}\,\underleftarrow{F}$ and on $\underleftarrow{\lim}F$ all have shadowing simultaneously. In particular, $F$ has shadowing if and only if the shift map on its directly induced generalized inverse limit $\underleftarrow{\lim}F$ has shadowing. This strengthens a recent theorem established under continuity and openness assumptions. We show that if the shift map on the generalized inverse limit is transitive (resp. weakly mixing, mixing, chain transitive, chain mixing), then the set-valued map is transitive (resp. weakly mixing, mixing, chain transitive, chain mixing). For a set-valued map with shadowing, the properties of total transitivity, weak mixing, mixing, specification and chain mixing are mutually equivalent.
