New York Journal of Mathematics
Volume 22 (2016) 943-988


Robin J. Deeley, D. Brady Killough, and Michael F. Whittaker

Dynamical correspondences for Smale spaces

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Published: September 15, 2016
Keywords: Smale space, hyperbolic dynamical system, generalized morphism
Subject: Primary 37D20; Secondary 37B10

We initiate the study of correspondences for Smale spaces. Correspondences are shown to provide a notion of a generalized morphism between Smale spaces and are a special case of finite equivalences. Furthermore, for shifts of finite type, a correspondence is related to a matrix which intertwines the adjacency matrices of the shifts. This observation allows us to define an equivalence relation on all Smale spaces which reduces to shift equivalence for shifts of finite type. Several other notions of equivalence are introduced on both correspondences and Smale spaces; a hierarchy between these equivalences is established. Finally, we provide several methods for constructing correspondences and provide specific examples.


This research was supported by the ANR Project SingStar and the Australian Research Council (DP130100490).

Author information

Robin J. Deeley:
Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Keller 401A, Honolulu HI, USA 96822

D. Brady Killough:
Department of Mathematics and Computing, Mount Royal University, Calgary, Alberta, Canada T3E 6K6

Michael F. Whittaker:
School of Mathematics and Statistics, University of Glasgow, Glasgow, Scotland Q12 8QW