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Covering action on Conley index theory

Part of: Topological dynamics Low-dimensional topology Birational geometry Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  21 February 2022

D. V. S. LIMA ,
M. R. DA SILVEIRA  and
E. R. VIEIRA
D. V. S. LIMA*
Affiliation:
CMCC, Federal University of ABC, Santo André, SP, Brazil (e-mail: mariana.silveira@ufabc.edu.br)
M. R. DA SILVEIRA
Affiliation:
CMCC, Federal University of ABC, Santo André, SP, Brazil (e-mail: mariana.silveira@ufabc.edu.br)
E. R. VIEIRA
Affiliation:
DIMACS, Rutgers University, Piscataway, NJ, USA (e-mail: ewerton.v@rutgers.edu) IME, Federal University of Goiás, Goiânia, GO, Brazil
*
e-mail: dahisy.lima@ufabc.edu.br
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Abstract

In this paper we apply Conley index theory in a covering space of an invariant set S, possibly not isolated, in order to describe the dynamics in S. More specifically, we consider the action of the covering translation group in order to define a topological separation of S which distinguishes the connections between the Morse sets within a Morse decomposition of S. The theory developed herein generalizes the classical connection matrix theory, since one obtains enriched information on the connection maps for non-isolated invariant sets, as well as for isolated invariant sets. Moreover, in the case of an infinite cyclic covering induced by a circle-valued Morse function, one proves that the Novikov differential of f is a particular case of the p-connection matrix defined herein.

Keywords

Conley indexconnection matrixcovering spacecircle-valued functionNovikov complex

MSC classification

Primary: 37B30: Index theory, Morse-Conley indices 14E20: Coverings
Secondary: 57M10: Covering spaces 58E40: Group actions 37B35: Gradient-like and recurrent behavior; isolated (locally maximal) invariant sets
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 5 , May 2023 , pp. 1633 - 1665
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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