时间重参数化下的时间度量等价和维数变化
Time-metric equivalence and dimension change under time reparameterizations
摘要：
我们研究了动力系统在时间参数化下的行为，这不仅对描述相对论系统中的混沌，而且对探索动力学量的不变性都很重要。我们首先表明，时间变换是局部等价的度量变换，结果导致任意黎曼相空间上的所有李雅普诺夫指数的变换规则。然后，我们表明，时间变换保持谱的广义尺寸D_q的信息维D_1，有趣的是，尽管以前的断言的不变性，在一个非平凡的方式转换除外。q=1时的不连续行为可以用来约束和扩展Kaplan-Yorke猜想的公式。
英文摘要：
We study the behavior of dynamical systems under time reparameterizations, which is important not only to characterize chaos in relativistic systems but also to probe the invariance of dynamical quantities. We first show that time transformations are locally equivalent to metric transformations, a result that leads to a transformation rule for all Lyapunov exponents on arbitrary Riemannian phase spaces. We then show that time transformations preserve the spectrum of generalized dimensions D_q except for the information dimension D_1, which, interestingly, transforms in a nontrivial way despite previous assertions of invariance. The discontinuous behavior at q=1 can be used to constrain and extend the formulation of the Kaplan-Yorke conjecture.
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