Resolving Extensions of Finitely Presented Systems
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Resolving Extensions of Finitely Presented Systems Todd Fisher
Received: 1 October 2009 / Accepted: 19 April 2010 / Published online: 6 March 2013 © Springer Science+Business Media Dordrecht 2013
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. Keywords Finitely presented · Resolving · Smale space Mathematics Subject Classification 37C15 · 37D05 · 37B99
1 Introduction One cornerstone of the study of dynamical systems is the theory of hyperbolic dynamics introduced by Smale and Anosov in the 1960s. For compact spaces the property of hyperbolicity, in general, produces highly nontrivial and interesting dynamics. The best understood hyperbolic sets are those that are locally maximal (or isolated). For many years it was asked if every hyperbolic set can be contained in a locally maximal hyperbolic set: in [7] it is shown that this is not the case for any manifold with dimension greater than one. This paper is in part an investigation into hyperbolic sets that are not locally maximal. If a hyperbolic set is not locally maximal one of the next properties one considers is the existence of a Markov partition, which is roughly a decomposition of the hyperbolic set into dynamically defined rectangles. In [7] it is shown that any hyperbolic set can be extended to one with a Markov partition.
Dedicated to the memory of Ki Hang Kim. T. Fisher () Department of Mathematics, Brigham Young University, Provo, UT 84602, USA e-mail: [email protected]
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The essential topological structure of a locally maximal hyperbolic set is captured with the notion of a Smale space, introduced by Ruelle [14] and simplified by Fried [8]. A Smale space is an expansive system with canonical coordinates (or a local product structure). Fried defined a finitely presented dynamical system as an expansive system which is a factor of a shift of finite type. The finitely presented systems contain the Smale spaces and share a great deal of their structure; in particular, they are precisely the expansive systems that admit Markov partitions [8]. In this paper we extend certain central results of zero dimensional finitely presented systems (sofic shifts) to higher dimensional finitely presented systems. This investigation is motivated in part by hyperbolic sets that need not be locally maximal, and also as a generalization to higher dimension of the symbolic viewpoint. More specifically, the present work looks at resolving maps from Smale spaces to finitely presented systems; these are absolutely central to the zero-dimensional theory. A factor map from
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