Globally self-similar metrics of ruelle expanding maps and global rigidity

The use of metric measure geometry in the study of dynamical systems has been an active and attractive area of research since several decades. In this paper, we will define and study the (length) self-similar metrics associated with Ruelle expanding maps, which are comparable to Hamenstädt metrics and Tits metrics associated with non-positively curved Riemannian manifolds. One of our results is as follows: Let M be a \(C^\infty \) $C^{\infty }$ compact connected manifold endowed with a length metric defining the manifold topology and \(\varphi \) $\phi$ be a \(C^\infty \) $C^{\infty }$ Ruelle expanding map of M. We will prove that if the length self-similar metric associated with \((M, \varphi )\) $(M,\phi )$ is induced from a Carnot-Carathéodory structure, then up to finite covers, \(\varphi \) $\phi$ is \(C^\infty \) $C^{\infty }$ conjugate to an expanding nilendomorphism.