Resolving Extensions of Finitely Presented Systems

Abstract

Abstract. In this paper we extend certain central results of zero dimensional systems to higher dimensions. The first main result shows that if (Y, f ) is a finitely presented system, then there exists a Smale space (X, F ) and a u-resolving factor map π + : X → Y . If the finitely presented system is transitive, then we show there is a canonical minimal u-resolving Smale space extension. Additionally, we show that any finite-to-one factor map between transitive finitely presented systems lifts through u-resolving maps to an s-resolving map.

The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors

“…This definition is adapted from a property pointed out by Bowen [8, p.13] for surjective continuous factor maps from shifts of finite type to systems associated with Markov partitions. More precisely, these factors are David Fried's finitely presented dynamical systems [18,19]; these are the expansive systems which are continuous factors of shifts of finite type.…”

Homology for one-dimensional solenoids

“…Smale spaces were defined by David Ruelle as a purely topological version of the basic sets of Axiom A systems which arise in Smale's program for differentiable dynamics [9,10,1,5,4]. Informally, a pair (X, ϕ), where X is a compact metric space and ϕ a homeomorphism of X, is a Smale space if it possesses local coordinates in contracting and expanding directions.…”

Order on the homology groups of Smale spaces

“…The original notion of a Smale space is due to David Ruelle, based on the observation that the basic sets of Smale's Axiom A systems do not form submanifolds of the ambient manifold [16,17,2,9,8]. In fact, Smale spaces are the topological dynamical systems that admit a hyperbolic structure in terms of canonical coordinates of contracting and expanding (or stable and unstable) directions.…”