Published as: Franks, John, and David Richeson. Shift Equivalence and the Conley Index. Transactions of the American Mathematical Society 352, no. 7 (2000): 3305-3322. https://www.ams.org/journals/tran/2000-352-07/S0002-9947-00-02488-0/home.html © 2000 American Mathematical Society This author post-print is made available on Dickinson Scholar with the permission of the publisher. For more information on the published version, visit American Mathematical Society’s (AMS) Website. © 2000. This publication is made available under the CC-BY-NC-ND 4.0 license: http://creativecommons.org/licenses/by-nc-nd/4.0/

In this paper we introduce filtration pairs for an isolated invariant set of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Last, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.