连续统意义下的双曲性
Continuum-wise hyperbolicity
AI总结：
本文提出连续统意义下的双曲性概念，证明cw-双曲同胚的阴影引理与谱分解，引入cwN-双曲性并研究其周期点有限性等动力学性质，还证明由环面拓扑双曲同胚诱导的球面同胚是cw-双曲且拓扑共轭于线性cw-Anosov微分同胚。
摘要：
我们引入连续统意义下的双曲性，这是相对于连续统理论对双曲性的一种推广。我们讨论拓扑双曲性与连续统意义下双曲性之间的相似与差异。针对cw-双曲同胚，以L-阴影性质的形式证明了一个阴影引理，并在该情形下得到了谱分解。在证明中，我们仅利用cw-扩张性推广了Fathi（文献[Fat89]）构造双曲度量的方法，得到一个双曲cw-度量。我们还引入cwN-双曲性，对任意大的$N\in\mathbb{N}$给出这些系统的例子，并获得这些系统的进一步动力学性质，例如具有相同周期的周期点的有限性。我们证明，由$\mathbb{T}^2$的拓扑双曲同胚所诱导的$\mathbb{S}^2$同胚是连续统意义下双曲的，并且拓扑共轭于$\mathbb{S}^2$上的线性cw-Anosov微分同胚，特别地是cw2-双曲的。
英文摘要：
We introduce continuum-wise hyperbolicity, a generalization of hyperbolicity with respect to the continuum theory. We discuss similarities and differences between topological hyperbolicity and continuum-wise hyperbolicity. A shadowing lemma for cw-hyperbolic homeomorphisms is proved in the form of the L-shadowing property and a Spectral Decomposition is obtained in this scenario. In the proof we generalize the construction of Fathi \cite{Fat89} of a hyperbolic metric using only cw-expansivity, obtaining a hyperbolic cw-metric. We also introduce cwN-hyperbolicity, exhibit examples of these systems for arbitrarily large $N\in\mathbb{N}$ and obtain further dynamical properties of these systems such as finiteness of periodic points with the same period. We prove that homeomorphisms of $\mathbb{S}^2$ that are induced by topologically hyperbolic homeomorphisms of $\mathbb{T}^2$ are continuum-wise-hyperbolic and topologically conjugate to linear cw-Anosov diffeomorphisms of $\mathbb{S}^2$, being in particular cw2-hyperbolic.
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