UDC 681.5+513.6+517.9
FORMAL AND NONARCHIMEDIAN STRUCTURES OF DYNAMIC SYSTEMS
ON MANIFOLDS
Abstract. New results are presented and a brief review of new methods and results of the theory of dynamic
systems on manifolds over local fields and formal groups over local rings is given. For the analysis of n-dimensional
manifolds and their dynamics, dynamic systems on such manifolds, formal structures are used, in particular, n-dimensional formal groups.
Infinitesimal deformations are presented in terms of formal groups.
The well-known one-dimensional case extends, and n-dimensional (n ≥1) analytic mappings
of an open p-adic polydisc (n-disk) Dpn are considered.
We introduce and investigate the n-dimensional analogs of modules arising in formal and non-Archimedean
dynamic structures. Attention is drawn to rigid structures, objects and methods. From the point of view of system analysis, new, namely,
formal and non-Archimedean, faces and structures of systems, maps and iterations of mappings between these faces and structures are introduced and investigated.
Keywords: formal group, local ring, commutative formal group scheme, deformation, formal module, dynamic system, module of differentials.
FULL TEXT
REFERENCES
- Krivonos Yu.G., Kharchenko V.P., Glazunov N.M. Differential-algebraic equations and dynamical systems on manifolds. Kibernetika i sistemnyj analiz. 2016. Vol. 52, No. 3. P. 83–96.
- Zerz E. Algebraic systems theory. Aachen: Lehrstuhl D fur Mathematik RWTH, 2006. 104 p.
- Hannan E.J., Deistler M. The statistical theory of linear systems. Philadelphia: SIAM Publ., 2012. 390 p.
- Wood, J. Modules and behaviours in nD systems theory. Multidimensional Systems and Signal Processing. 2000. Vol. 11, Iss. 1–2. P. 11–48.
- Arbib MA, Mains E., Brockett R., Lobri K., Burns K.I., Hart N.E., Ossetian N.I. Mathematical methods in systems theory (Russian translation). Moscow: Mir, 1979. 328 p.
- Systems Theory. Mathematical methods and modeling (in Russian). Moscow: Mir, 1989. 382 p.
- Glushkov V.M. Automata theory and formal firmware transformations. Kibernetika. 1965. No. 5. P. 1–10.
- Glushkov V.M. Introduction to ACS (in Russian). Kiev: Technika, 1974. 320 p.
- Glazunov M.M. On the "normal subgroup" of one-dimensional formal groups defined over a ring of entire local fields. Dop. Akad. Nauk URSR. Ser. А. 1973. No. 11. P. 965–968.
- Lubin J. Non-archimedean dynamical systems. Compos. Math. 1994. Vol. 94. P. 321–346.
- Hua-Chien Li. p-adic dynamical systems and formal groups. Compos. Math. 1996. Vol. 104. P. 41–54.
- Serre J.P. Lie algebras and Lie groups. Lecture Notes in Mathematics. Vol. 1500. Berlin; Heidelberg: Springer, 1992. 168 p.
- Pontryagin L.S. Continuous groups: 3rd ed. Moscow: Nauka, 1986. 519 p.
- Hazewinkel M. Formal groups and applications, Providence, Rhode Island: AMS Chelsea Publishing, 2012. 573 p.
- Mumford D. Abelian varieties. Tata Institute of fundamental research publications, Vol. 13, 2012. 263 p.
- Schwede S. Equivariant properties of symmetric products. J. Amer. Math. Soc. 2017. Vol. 30, N 3. P. 673–711.
- Schwede S. Formal groups and stable homotopy of commutative rings. Geometry & Topology. 2004. Vol. 8. P. 335–412.
- Snaith V. Stable homotopy around the Arf-Kervaire invariant. Basel; Boston; Berlin: Birkhauser, 2009. 240 p.
- Faltings G. p–adic hodge theory. J. Amer. Math. Soc. 1988. Vol. 1, N 1. P. 255–288.
- Kisin M. Crystalline representations and F-crystals. In: Algebraic Geometry and Number Theory. Progress in Mathematics. Ginzburg V. (ed.). Boston: Birkhuser, 2006. Vol. 253. P. 459–496.
- Glazunov N.M. Extremal forms and rigidity in arithmetic geometry and in dynamics. Chebyshevskii Sbornik. 2015. Vol. XVI, Iss. 3(55). P. 124–146.
- Glazunov N.M. On norm maps and “universal norms” of formal groups over integer rings of local fields. Continuous and Distributed Systems. Theory and Applications. Berlin; Heidelberg: Springer, 2014. P. 73–80.
- Khrennikov A.Yu., Nilsson M. p-adic deterministic and random dynamics. Dordrecht: Kluwer Academic Publ, 2004. 280 p.
- Vladimirov V.S., Volovich I.V., Zelenov E.I. p-adic analysis and mathematical physics. Series on Soviet and East European Mathematics. N.Y.: World Scientific Co., Inc. 1994. Vol. 1. 340 p.
- Woodcock C.F., Smart N.P. p-adic chaos and random number generation. Experiment Math. 1998. P. 333–342.
- Thiran E., Verstegen D., Weyers J. p-adic dynamics. J. Stat. Phys. 1989. Vol. 54. P. 893–913.
- Ben-Menahem S. p-adic iterations. Preprint, TAUP 1627–88, Tel Aviv University, 1988. 43 p.
- Gromov M. Soft and hard symplectic geometry. Proc. of the International Congress of Mathematicians Berkeley. California, USA, 1986. P. 81–98.
- Postnikov A.G. Selected Works. Moscow: Fizmatlit, 2005. 512 p.
- Rigidity (mathematics). URL: https://en.wikipedia.org/wiki/Rigidity_(mathematics).
- Serre J.P. Corps locaux. Paris: Hermann. 2004. 246 p.
- Field M., Jarden M. Field arithmetic. Third ed. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 11, Berlin: Springer-Verlag, 2008. 792 p.
- Shafarevich I.R. Mathematical works. V. 3, Part 1. Moscow: Prima B., 1996. 415 p.
