AI 中文总结
本文研究一类含滑动Shilnikov连接附近首次回归映射的一维分段扩张映射，借助共形迭代函数系统理论，证明其不变集上动力学与$\be{N}^{\be{N}}$上移位拓扑共轭，存在唯一不变遍历共形测度。
AI 中文摘要
本文研究一类由无穷多个光滑扩张分支定义的一维分段映射，这类映射自然产生于非光滑动力系统语境，包含在滑动Shilnikov连接附近局部定义的首次回归映射。借助共形迭代函数系统（CIFS）理论，我们探究这些映射的若干动力性质及其不变集的拓扑复杂性。特别地，我们证明限制在不变集上的动力学在拓扑上与$\boldsymbol{\be{N}}^{\boldsymbol{\be{N}}}$上的移位拓扑共轭，还建立了唯一共形测度的存在性，该测度在映射下不变且具有遍历性。
英文摘要
In this paper, we study a class of one-dimensional piecewise maps defined by infinitely many smooth expanding branches. This class arises naturally in the context of non-smooth dynamical systems and includes the first-return maps locally defined near sliding Shilnikov connections. By means of the theory of conformal iterated function systems (CIFS), we investigate several dynamical properties of these maps as well as the topological complexity of their invariant sets. In particular, we show that the dynamics restricted to the invariant set is topologically conjugate to the shift on $\mathbb{N}^{\mathbb{N}}$. We also establish the existence of a unique conformal measure that is invariant and ergodic under the map.
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