AI 中文总结
该研究分析Schwartz空间上复合算子的线性动力学，证明可逆复合算子非广义双曲、符号有不动点时算子无正阴影性质，给出符号使算子正拓扑膨胀的条件，刻画仿射符号并明确奇偶次多项式符号的膨胀性差异。
AI 中文摘要
在本注记中，我们证明S(R)上的可逆复合算子绝不是广义双曲的；当符号具有不动点时，对应算子不具有正阴影性质。随后，我们建立符号满足的充分条件，使得关联复合算子是正拓扑膨胀的。特别地，我们对仿射符号得到完全刻画，证明广泛类奇次多项式符号的膨胀性，同时表明偶次多项式符号无法生成拓扑膨胀算子。我们的结果揭示了符号动力学与诱导复合算子线性动力学之间的强关联，为局部凸空间中的拓扑膨胀性提供了新例子与阻碍。
英文摘要
In this note, we show that invertible composition operators on S(R) are never generalized hyperbolic and, when the symbol has a fixed point, the corresponding operator fails to have the positive shadowing property. We then establish sufficient conditions on the symbol under which the associated composition operator is positively topologically expansive. In particular, we obtain a complete characterization for affine symbols and prove expansivity for broad classes of odd-degree polynomial symbols, while showing that polynomial symbols of even degree cannot generate topologically expansive operators. Our results reveal a strong connection between the dynamics of the symbol and the linear dynamics of the induced composition operator, and provide new examples and obstructions for topological expansivity in locally convex spaces.
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