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.
