Linear Relations of Finite Length Modules are Shift Equivalent to Maps

• Bartosz Furmanek ^[1] ; Filip Oskar Łanecki ^[1] ; Mateusz Przybylski ^[1] ; Jim Wiseman ^[2]
  1. [1] Jagiellonian University

     Jagiellonian University

     Kraków, Polonia

     
  2. [2] Agnes Scott College

     Agnes Scott College

     Estados Unidos

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 3, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Linear relations, defined as submodules of the direct sum of two modules, can be viewed as objects that carry dynamical information and reflect the inherent uncertainty of sampled dynamics. These objects also provide an algebraic structure that enables the definition of subtle invariants for dynamical systems. By recognizing specific categorical aspects of these objects, we prove that linear relations defined on modules of finite length are shift equivalent to bijective mappings.

• Referencias bibliográficas
  • 1. Akin, E., Mrozek, M., Przybylski, M., Wiseman, J.: A complete invariant for shift equivalence for Boolean matrices and finite relations....
  • 2. Barmak, J., Mrozek, M., Wanner, T.: Conley index for multivalued maps on finite topological spaces. Found. Comput. Math. 26, 647–687 (2024)....
  • 3. Batko, B.,Gameiro, M.,Hung,Y.,Kalies,W., Mischaikow, K.,Vieira, E.: Identifying nonlinear dynamics with high confidence from sparse data....
  • 4. Batko, B.,Mischaikow, K., Mrozek,M., Przybylski, M.: Conley index approach to sampled dynamics. SIAM J. Appl. Dyn. Syst. 19(1), 665–704...
  • 5. Chen, C.-C., Conejero, J.A., Kosti´c, M.,Murillo-Arcila, M.: Dynamics of multivalued linear operators. Open Math. 15(1), 948–958 (2017)
  • 6. Conley, C.: Isolated Invariant Sets and The Morse Index. Regional conference series in mathematics, no. 38. AMS (1978)
  • 7. Franks, J., Richeson, D.: Shift equivalence and the Conley index. Trans. Am. Math. Soc. 352(7), 3305–3322 (2000)
  • 8. Juda, M.: Unsupervised features learning for sampled vector fields. SIAM J. Appl. Dyn. Syst. 19(4), 2720–2736 (2020)
  • 9. Kaczynski, T., Mrozek, M.: Conley index for discrete multi-valued dynamical systems. Topol. Appl. 65(1), 83–96 (1995)
  • 10. Mac Lane, S.: Homology. Classics in Mathematics. Springer Berlin, Heidelberg (1995)
  • 11. Moore, R.E.: Interval analysis. Series in automatic computation, Prentice-Hall (1966)
  • 12. Mrozek, M.: Leray functor and cohomological Conley index for discrete dynamical systems. Trans. Am. Math. Soc. 318(1), 149–178 (1990)
  • 13. Szymczak, A.: The Conley index for discrete semidynamical systems. Topol. Appl. 66(3), 215–240 (1995)
  • 14. Mrozek,M.: Construction and properties of theConley index. Banach Center Publ. 47(1), 29–40 (1999)
  • 15. Przybylski, M.,Mrozek,M.,Wiseman, J.: The Szymczak functor and shift equivalence on the category of finite sets and finite relations....
  • 16. Robbin, J.W., Salamon, D.: Dynamical systems, shape theory and the Conley index. Ergodic Theory Dyn. Syst. 8, 375–393 (1988)
  • 17. Vietoris, L.: Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen. Math. Ann. 97(1), 454–472...