Lifting factor maps to resolving maps (Q1781941)

The paper considers homeomorphisms of compact metric spaces with dense orbits, every point nonwandering, and, roughly speaking, a hyperbolic structure (the actual hypothesis is the existence of local canonical coordinates determined by stable and unstable directions). Such a system is called an irreducible Smale space. The main result is that if there is a factor map \(h\) between two such systems such that for some point \(y\), the inverse image \(h^{-1}(y)\) consists of a single point, then the factor system \(h:(X,f)\rightarrow (Y,g)\) lifts to an another factor system \(h':(X',f')\rightarrow (Y',g')\) where \(h'\) is injective on local stable sets of \((X',f')\). The lifting involves factor maps \(\alpha:(X',f')\rightarrow (X,f)\) and \(\beta:(Y',g)\rightarrow (Y,g)\) that are injective on local unstable sets and such that \(\beta\circ h' = h\circ\alpha\). As a corollary, it is shown that given an irreducible Smale space \((Y,g)\) then there are two others \((X,f)\) and \((\Sigma,\sigma)\) and factor maps \(h_1:(\Sigma,\sigma)\rightarrow (X,f)\) and \(h_2:(X,f)\rightarrow (Y,g)\) with \((\Sigma,\sigma)\) a subshift of finite type, \(h_1\) injective on local stable sets, and \(h_2\) injective on local unstable sets.