Optimal Lyapunov metrics of expansive homeomorphisms

We sharpen the following results of Reddy, Sakai and Fried: any expansive homeomorphism of a metrizable compactum admits a Lyapunov metric compatible with the topology, and if we also assume the existence of a local product structure (that is, if the homeomorphism is an A$^{\#}$-homeomorphism in the terminology of Alekseev and Yakobson, or possesses hyperbolic canonical coordinates in the terminology of Bowen, or together with the metric compactum constitutes a Smale space in the terminology by Ruelle), then we also obtain the validity of Ruelle's technical axiom on the Lipschitz property of the homeomorphism, its inverse, and the local product structure. It is shown that any expansive homeomorphism admits a Lyapunov metric such that the homeomorphism on local stable (resp. unstable) “manifolds” is approximately representable on a small scale as a contraction (resp. expansion) with constant coefficient $\lambda_s$ (resp. $\lambda_u^{-1}$) in this metric. For A$^{\#}$-homeomorphisms, we prove that the desired metric can be approximately represented on a small scale as the direct sum of metrics corresponding to the canonical coordinates determined by the local product structure and that local “manifolds” are “flat” in some sense. It is also proved that the lower bounds for the contraction constants $\lambda_s$ and expansion constants $\lambda_u$ of A$^{\#}$-homeomorphisms are attained simultaneously for some metric that satisfies all the conditions described.