AI 中文总结
本文在拓扑传递假设下研究Lipschitz自由空间上线性化算子的性质，证明了连通空间上Lipschitz算子的Kitai定理，即复变体每个连通分量与单位圆相交。
AI 中文摘要
Lipschitz自由空间的泛性质允许将保基点Lipschitz映射$f\colon M\to M$线性化为有界线性算子$T_f\colon \mathcal{F}(M) \to \mathcal{F}(M)$。已知当$f$具有弱混合性质时，算子$T_f$也是弱混合的，但拓扑传递性是否具有类似的继承结果仍是开放问题。受此问题启发，我们在$f$拓扑传递的假设下研究$T_f$的性质。特别地，我们证明了连通空间上Lipschitz算子的Kitai定理，即若$f$拓扑传递，则$T_f$的复变体的每个连通分量都与单位圆相交。
英文摘要
The universal property of the Lipschitz-free spaces allows for the linearisation of a base point preserving Lipschitz map $f\colon M\to M$ to a bounded linear operator $T_f\colon \mathcal{F}(M) \to \mathcal{F}(M)$. While it is known that the operator $T_f$ is weakly mixing whenever $f$ has this property, it is open whether a similar inheritance result for topological transitivity holds. Motivated by this question we investigate properties of $T_f$ under the assumption that $f$ is topologically transitive. In particular we prove Kitai's theorem for Lipschitz operators, i.e. we show that every connected component of the complex variant of $T_f$ intersects the unit circle if $f$ is topologically transitive.
Comments14 pages; Generalised Theorem 3.7 to the case of not-necessarily connected spaces
