We address the problem of defining Lyapunov exponents for an expansive homeomorphism f on a compact metric space (X, dist) using similar techniques as those developed in Barreira and Silva [Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dynam. Sys.13 (2005), 469–490]; Kifer [Characteristic exponents of dynamical systems in metric spaces, Ergod. Th. Dynam. Sys.3 (1983), 119–127]. Under certain conditions on the topology of the space X where f acts we obtain that there is a metric D defining the topology of X such that the Lyapunov exponents of f are different from zero with respect to D for every point x ∈ X. We give an example showing that this may not be true with respect to the original metric dist. But expansiveness of f ensures that Lyapunov exponents do not vanish on a Gδ subset of X with respect to any metric defining the topology of X. We define Lyapunov exponents on compact invariant sets of Peano spaces and prove that if the maximal exponent on the compact set is negative then the compact is an attractor.

中文翻译：

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扩展同胚的 Lyapunov 指数

我们解决了为扩展同胚定义 Lyapunov 指数的问题F在紧度量空间上 (X,距离) 使用与 Barreira 和 Silva [Lyapunov exponents for continuous transformations and dimension theory,离散康定。动态。系统。13(2005), 469–490]；Kifer [度量空间中动力系统的特征指数，埃尔戈德。钍。动态。系统。3(1983), 119–127]。在一定条件下的空间拓扑X在哪里F我们获得有度量的行为D定义拓扑X使得 Lyapunov 指数F与零不同D对于每一点X∈X. 我们举了一个例子，表明这对于原始度量 dist 可能不正确。但扩展性F确保 Lyapunov 指数不会在Gδ的子集X关于定义拓扑的任何度量X. 我们在 Peano 空间的紧不变集上定义 Lyapunov 指数，并证明如果紧集上的最大指数为负，则紧集是吸引子。