Abstract
In this paper, we study the behavior of time averages of the measure in Bowan’s example: a vector field on the plane with two saddles joined by two separatrix connections. We present an explicit criterion for the convergence of averaged measures and describe the set of their partial limits. As a consequence, we show that, for a typical initial measure, its time averages do not converge.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Gaunersdorfer, “Time averages for heteroclinic attractors,” SIAM J. Appl. Math. 52(5), 1476–1489 (1992).
F. Takens, “Heteroclinic Attractors: time averages and moduli ofpological conjugacy,” Bol. Soc. Brasil. Mat. (N. S.) 25(1), 107–120 (1994).
T. Young, “Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure,” Discrete Contin. Dyn. Syst. 9(2), 359–378 (2003).
Ya. G. Sinai, “Gibbs measures in ergodic theory,” Uspekhi Mat. Nauk 27(4), 21–64 (1972).
D. Ruelle, “A measure associated with axiom-A attractors,” Amer. J. Math. 98(3), 619–654 (1976).
R. Bowen, Equilibrium States and Ergodic Theory of Anosov Diffeomorphisms, in Lecture Notes in Math. (Springer, Berlin-New York, 1975), Vol. 470.
M. Blank and L. Bunimovich, “Multicomponent dynamical systems: SRB measures and phase transitions,” Nonlinearity 16(1), 387–401 (2003).
M. Misiurewicz, Ergodic Natural Measures, Preprint (2004).
E. Järvenpää and T. Tolonen, “Relations between natural and observable measures,” Nonlinearity 18(2), 897–912 (2005).
L.-S. Young, “What are SRB measures and which dynamical systems have them?,” J. Stat. Phys. 108(5–6), 733–754 (2002).
M. Misiurewicz and A. Zdunik, “Convergence of images of certain measures,” in Progr. Phys., Vol. 10: Statistical Physics and Dynamical Systems, Közeg, 1984 (Birkhäuser Boston, Boston, MA, 1985), pp. 203–219.
V. Kleptsyn, “An example of non-coincidence of minimal and statistical attractors,” Ergodic Theory Dynam. Systems 26(3), 759–768 (2006).
Yu. S. Il’yashenko, “Dulac’s memoir “On limit cycles” and related questions of the local theory of differential equations,” Uspekhi Mat. Nauk 40(6), 41–78 (1985).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © T. Golenishcheva-Kutuzova, V. Kleptsyn, 2007, published in Matematicheskie Zametki, 2007, Vol. 82, No. 5, pp. 678–689.
Rights and permissions
About this article
Cite this article
Golenishcheva-Kutuzova, T., Kleptsyn, V. Convergence of the Krylov-Bogolyubov procedure in Bowan’s example. Math Notes 82, 608–618 (2007). https://doi.org/10.1134/S0001434607110041
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0001434607110041