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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Detection of renewal system factors via the Conley index


Author: Jim Wiseman
Journal: Trans. Amer. Math. Soc. 354 (2002), 4953-4968
MSC (2000): Primary 37B30; Secondary 37B10, 54H20
Published electronically: August 1, 2002
MathSciNet review: 1926844
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Abstract: Let $N$ be an isolating neighborhood for a map $f$. If we can decompose $N$ into the disjoint union of compact sets $N_1$ and $N_2$, then we can relate the dynamics on the maximal invariant set $\operatorname{Inv} N$ to the shift on two symbols by noting which component of $N$ each iterate of a point $x\in \operatorname{Inv} N$ lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to $N$. In essence, we measure the difference between the Conley index of $\operatorname{Inv}N$and the sum of the indices of $\operatorname{Inv} N_1$ and $\operatorname{Inv} N_2$.


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Additional Information

Jim Wiseman
Affiliation: Northwestern University, Evanston, Illinois 60208
Address at time of publication: Department of Mathematics and Statistics, Swarthmore College, 500 College Ave., Swarthmore, Pennsylvania 19081
Email: jwisema1@swarthmore.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03063-5
PII: S 0002-9947(02)03063-5
Keywords: Conley index, symbolic dynamics, renewal system
Received by editor(s): August 5, 2001
Received by editor(s) in revised form: March 28, 2002
Published electronically: August 1, 2002
Article copyright: © Copyright 2002 American Mathematical Society