Iterated Means Dichotomy for Discrete Dynamical Systems

• Saburov Mansoor ^[1]
  1. [1] American University of the Middle East
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00379-0
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we discuss a dichotomy of iterated means for a compact discrete dynamical system acting on a finite dimensional space. As an application, we also study the mean ergodicity of non-homogeneous Markov chains.

• Referencias bibliográficas
  • 1. Baillon, J.B.: Un theoreme de type ergodique pour les contractions non lineaires dans un espace de Hilbert. C. R. Acad. Sci. Paris 280,...
  • 2. Baranski, K., Misiurewicz, M.: Omega-limit sets for the Stein–Ulam spiral map. Top. Proc. 36, 145–172 (2010)
  • 3. Bernstein, S.: Solution of a mathematical problem connected with the theory of heredity. Ann. Math. Stat. 13, 53–61 (1942)
  • 4. Eisner, T., Farkas, B., Haase, M., Nagel, R.: Operator Theoretic Aspects of Ergodic Theory. Springer, New York (2015)
  • 5. Ganikhodjaev, N., Ganikhodjaev, R., Jamilov, U.: Quadratic stochastic operators and zero-sum game dynamics. Ergod. Theory Dyn. Syst. 35(5),...
  • 6. Ganikhodzhaev, N.N., Zanin, D.V.: On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex....
  • 7. Hardy, G.H.: Theorems relating to the summability and convergence of slowly oscillating series. Proc. Lond. Math. Soc. s2–8, 301–320 (1910)
  • 8. Hardy, G.H.: Divergent Series. American Mathematical Society, Providence (2000)
  • 9. Kesten, H.: Quadratic transformations: a model for population growth. I. Adv. Appl. Probab. 2, 1–82 (1970)
  • 10. Kolokoltsov, V.: Nonlinear Markov Processes and Kinetic Equations. Cambridge University Press, New York (2010)
  • 11. Lyubich, Y.I.: Mathematical Structures in Population Genetics. Springer, Berlin (1992)
  • 12. Menzel, M.T., Stein, P.R., Ulam, S.M.: Quadratic Transformations, Part I, Vol. 2305. Los Alamos Scientific Laboratory of the University...
  • 13. Ruelle, D.: Historical behavior in smooth dynamical systems. In: Broer, H.W., et al. (eds.) Global Analysis of Dynamical Systems, pp....
  • 14. Saburov, M.: A class of nonergodic Lotka–Volterra operators. Math. Notes 97(5–6), 759–763 (2015)
  • 15. Saburov, M.: On divergence of any order Cesaro mean of Lotka–Volterra operators. Ann. Funct. Anal. 6(4), 247–254 (2015)
  • 16. Saburov, M.: Ergodicity of nonlinear Markov operators on the finite dimensional space. Nonlinear Anal. Theory Methods Appl. 143, 105–119...
  • 17. Saburov, M.: On the surjectivity of quadratic stochastic operators acting on the simplex. Math. Notes 99(4), 623–627 (2016)
  • 18. Saburov, M.: On regularity of diagonally positive quadratic doubly stochastic operators. Results Math. 72, 1907–1918 (2017)
  • 19. Saburov, M.: On regularity of positive quadratic doubly stochastic operators. Math. Notes 103(2), 328–333 (2018)
  • 20. Saburov, M.: Dichotomy of iterated means for nonlinear operators. Funct. Anal. Appl. 52(1), 89–91 (2018)
  • 21. Saburov, M.: Ergodicity of p-majorizing quadratic stochastic operators. Markov Process. Relat. Fields 24(1), 131–150 (2018)
  • 22. Saburov, M.: Ergodicity of p-majorizing nonlinear Markov operators on the finite dimensional space. Linear Algebra Appl. 578, 53–74 (2019)
  • 23. Saburov, M., Saburov, Kh: Reaching a consensus in multi-agent systems: a time invariant nonlinear rule. J. Educ. Vocat. Res. 4(5), 130–133...
  • 24. Saburov, M., Saburov, Kh: Mathematical models of nonlinear uniform consensus. ScienceAsia 40(4), 306–312 (2014)
  • 25. Saburov, M., Saburov, Kh: Reaching a nonlinear consensus: polynomial stochastic operators. Int. J. Control Autom. Syst. 12(6), 1276–1282...
  • 26. Saburov, M., Saburov, Kh: Reaching a nonlinear consensus: a discrete nonlinear time-varying case. Int. J. Syst. Sci. 47(10), 2449–2457...
  • 27. Saburov, M., Saburov, K.: Reaching consensus via polynomial stochastic operators: a general study. In: Springer Proceedings in Mathematics...
  • 28. Saburov, M., Yusof, N.A.: Counterexamples to the conjecture on stationary probability vectors of the second-order Markov chains. Linear...
  • 29. Seneta, E.: Nonnegative Matrices and Markov Chains. Springer, New York (1981)
  • 30. Ulam, S.: A Collection of Mathematical Problems. Interscience Publishers, New York (1960)
  • 31. Vallander, S.S.: The limiting behavior of the sequences of iterates of certain quadratic transformations. Sov. Math. Dokl. 13, 123–126...
  • 32. Zakharevich, M.: On the behavior of trajectories and the ergodic hypothesis for quadratic mappings of a simplex. Rus. Math. Surv. 33(6),...