A Cr unimodal map with an arbitrary fast growth of the number of periodic points

Article English OPEN
Kaloshin, V. ; Kozlovski, O. (2012)

In this paper we present a surprising example of a C(r) unimodal map of an interval f : I -> I whose number of periodic points P(n)(f) = vertical bar{x is an element of I : f(n) x = x}vertical bar grows faster than any ahead given sequence along a subsequence (n)k = 3(k). This example also shows that 'non-flatness' of critical points is necessary for the Martens de Melo van Strien theorem [M. Martens, W. de Melo and S. van Strien. Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math. 168(3-4) (1992), 273-318] to hold.
  • References (12)
    12 references, page 1 of 2

    M. Artin and B. Mazur. On periodic orbits. Ann. of Math. (2) 81 (1965), 82-99.

    H. Epstein. Fixed points of composition operators. Proceedings of NATO Advanced Study Institute on Nonlinear Evolution, Italy. Plenum, New York, 1988, pp. 71-100.

    Phys. D 62 (1993), 1-14.

    V. Kaloshin. An extension of the Artin-Mazur theorem. Ann. of Math. (2) 150 (1999), 729-741.

    Comm. Math. Phys. 211(1) (2000), 253-271.

    V. Kaloshin. Growth of the number of periodic points. Normal Forms, Bifurcations and Finiteness Problems in Differential Equations. Eds. Y. Ilyashenko and C. Rousseau. Kluwer, Dordrecht, 2004, pp. 355-385.

    V. Kaloshin and M. Saprykina. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete Contin. Dyn. Syst. 15(2) (2006), 611-640.

    O. S. Kozlovski. The dynamics of intersections of analytical manifolds. Dokl. Akad. Nauk 323(5) (1992), 823-825 (Engl. transl. Russian Acad. Sci. Dokl. Math. 45(2) (1992), 425-427).

    (N.S.) 6(3) (1982), 427-434.

    M. Martens. Periodic points of renormalization. Ann. of Math. (2) 147(3) (1998), 435-484.

  • Metrics
    No metrics available
Share - Bookmark