Hostname: page-component-669899f699-swprf Total loading time: 0 Render date: 2025-04-24T17:19:21.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Covering action on Conley index theory

Part of: Topological dynamics Birational geometry Low-dimensional topology 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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aparicio Monforte, A. and Kauers, M.. Formal Laurent series in several variables. Expo. Math. 31(4) (2013), 350–367.CrossRefGoogle ScholarPubMed
Banyaga, A. and Hurtubise, D.. Lectures on Morse Homology (Kluwer Texts in the Mathematical Sciences,29). Springer, Dodrecht, 2013.Google Scholar
Clark, A.. Solenoidalization and denjoids. Houston J.Math. 26(4) (2000), 661–692.Google Scholar
Conley, C.. Isolated Invariant Sets and the Morse Index (CBMSRegional Conference Series in Mathematics, 38). American Mathematical Society, Providence,RI, 1978.CrossRefGoogle Scholar
Cornea, O. and Ranicki, A.. Rigidity and gluing for Morse and Novikov complexes. J. Eur. Math. Soc.(JEMS) 5(4) (2003), 343–394.CrossRefGoogle Scholar
de S. Lima, D. V. and deRezende, K. A.. Connection matrices for Morse–Bott flows. Topol. Methods NonlinearAnal. 44(2) (2014), 471–495.CrossRefGoogle Scholar
de S. Lima, D. V., Neto, O. M., de Rezende, K. A. andda Silveira, M. R.. Cancellations for circle-valued Morsefunctions via spectral sequences. Topol. Methods Nonlinear Anal. 51(1)(2018), 259–311.Google Scholar
Franzosa, R.. Index filtrations and the homology index braid for partiallyordered Morse decompositions. Trans. Amer. Math. Soc. 298(1) (1986),193–213.CrossRefGoogle Scholar
Franzosa, R., deRezende, K. A. and Vieira, E. R.. Generalizedtopological transition matrix. Topol. Methods Nonlinear Anal. 48(1)(2016), 183–212.CrossRefGoogle Scholar
Franzosa, R. and Vieira, E. R.. Transition matrix theory. Trans. Amer. Math. Soc. 369(11) (2017), 7737–7764.CrossRefGoogle Scholar
Franzosa, R. D.. The connection matrix theory for Morsedecompositions. Trans. Amer. Math. Soc. 311(2) (1989),561–592.CrossRefGoogle Scholar
McCord, C.. The connection map for attractor-repeller pairs.Trans. Amer. Math. Soc. 307(1) (1988),195–203.CrossRefGoogle Scholar
McCord, C., Mischaikow, K. and Mrozek, M.. Zetafunctions, periodic trajectories, and the Conley index. J. Differential Equations 121(2) (1995), 258–292.CrossRefGoogle Scholar
Mischaikow, K. and Mrozek, M.. Isolating neighborhoods and chaos. Jpn. J. Ind. Appl. Math. 12(2) (1995), 205–236.CrossRefGoogle Scholar
Mischaikow, K. and Mrozek, M.. Conley Index (Handbook of Dynamical Systems, 2).North-Holland, Amsterdam, 2002, pp.393–460.Google Scholar
Mrozek, M. and Pilarczyk, P.. The Conley index and rigorous numerics for attracting periodic orbits.Variational and Topological Methods in the Study of Nonlinear Phenomena (Progress in Nonlinear Differential Equations and TheirApplications, 49). Eds. V. Benci, G. Cerami, M. Degiovanni, D. Fortunato, F. Giannoni and A. M. Micheletti. Birkhäuser, Boston, 2002, pp.65–74.CrossRefGoogle Scholar
Pajitnov, A. V.. Circle-Valued Morse Theory (De Gruyter Studies inMathematics, 32). Walter de Gruyter, Berlin, 2008.Google Scholar
Ranicki, A.. Circle valued Morse theory and Novikov homology.Topology of High-Dimensional Manifolds, Nos. 1, 2 (Trieste, 2001) (ICTP Lecture Notes, 9). Eds. F.T. Farrell, L. Goettshe and W. Lueck. Abdus Salam International Centre for Theoretical Physics, Trieste, 2002, pp.539–569.Google Scholar
Reineck, J. F.. The connection matrix in Morse–Smaleflows. Trans. Amer. Math. Soc. 322(2) (1990),523–545.CrossRefGoogle Scholar
Salamon, D.. Connected simple systems and the Conley index of isolatedinvariant sets. Trans. Amer. Math. Soc. 291(1) (1985),1–41.CrossRefGoogle Scholar
Salamon, D.. Morse theory, the Conley index and Floerhomology. Bull. Lond. Math. Soc. 22(2) (1990),113–140.Google Scholar
Weber, J.. The Morse–Witten complex via dynamicalsystems. Expo. Math. 24(2) (2006),127–159.CrossRefGoogle Scholar