复杂度准度量空间上的扩展同胚
Expansive homeomorphisms on complexity quasi-metric spaces
AI总结：
本文研究复杂度准度量空间上的扩展同胚理论，证明缩放变换在该空间上扩张当且仅当α≠1，并将轨道分离与经典时间层级定理联系起来。
摘要：
Schellekens的复杂度准度量是一种拓扑框架，将计算比较的不对称性内置于距离本身。本文发展了该空间上的扩展同胚理论。核心结果是缩放变换ψ_α(f)(n)=αf(n)在复杂度空间(C,d_C)上扩张当且仅当α≠1。该动态的δ-稳定集与渐进复杂度类一致，为复杂度理论中熟知的对象提供了动力学刻画。随后证明了ψ_α的坐标是双曲的，收缩率λ=1/α，并将动力系统中的轨道分离与Hartmanis和Stearns的经典时间层级定理联系起来。不稳定集、共轭动力学和缩放映射的拓扑熵估计也被研究。每项证明都配有具体算法和Python实现，SageMath片段与示例并列，完整代码见https://github.com/gabayae/expansive-homeomorphisms-complexity-qmetric。
英文摘要：
The complexity quasi-metric of Schellekens is a topological framework in which the asymmetry of computational comparisons -- ``$A$ is at most as fast as $B$'' carrying different information than ``$B$ is at most as slow as $A$'' -- is built into the distance itself. This paper develops the theory of expansive homeomorphisms on the resulting space. The central result is that the scaling transformation $ψ_α(f)(n)=αf(n)$ is expansive on the complexity space $(\C,d_\C)$ if and only if $α\neq 1$. The $δ$-stable sets of this dynamics turn out to coincide with asymptotic complexity classes, giving a dynamical characterisation of objects familiar from complexity theory. We then show that the canonical coordinates of $ψ_α$ are hyperbolic with contraction rate $λ=1/α$, and we connect orbit separation in the dynamical system to the classical time hierarchy theorem of Hartmanis and Stearns. Unstable sets, conjugate dynamics, and topological entropy estimates for the scaling map are also worked out. Concrete algorithms and Python implementations accompany every proof, so each result can be checked computationally; SageMath snippets sit alongside the examples, and the full code is in the \href{https://github.com/gabayae/expansive-homeomorphisms-complexity-qmetric}{companion repository}.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
