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1998 | 157 | 2-3 | 245-254
Tytuł artykułu

Entropy and growth of expanding periodic orbits for one-dimensional maps

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let f be a continuous map of the circle $S^1$ or the interval I into itself, piecewise $C^1$, piecewise monotone with finitely many intervals of monotonicity and having positive entropy h. For any ε > 0 we prove the existence of at least $e^{(h-ε)n_k}$ periodic points of period $n_k$ with large derivative along the period, $|(f^{n_k})'| > e^{(h-ε)n_k}$ for some subsequence ${n_k}$ of natural numbers. For a strictly monotone map f without critical points we show the existence of at least $(1-ε) e^{hn}$ such points.
Słowa kluczowe
Rocznik
Tom
157
Numer
2-3
Strony
245-254
Opis fizyczny
Daty
wydano
1998
otrzymano
1998-01-21
poprawiono
1998-03-02
Twórcy
autor
  • Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A., katok_a@math.psu.edu
autor
  • Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A., mezhirov@math.psu.edu
Bibliografia
  • [ALM] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Sci., Singapore, 1993.
  • [BKP] L. Barreira, A. Katok and Ya. Pesin, Non-Uniformly Hyperbolic Dynamical Systems, monograph in preparation.
  • [Bu] J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math. 100 (1997), 125-161.
  • [GS] P. Góra et B. Schmitt, Un exemple de transformation dilatante et $C^1$ par morceaux de l'intervalle, sans probabilité absolument continue invariante, Ergodic Theory Dynam. Systems 9 (1989), 101-113.
  • [Ho] F. Hofbauer, The structure of piecewise monotonic transformations, ibid. 1 (1981), 159-178.
  • [K1] A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphisms, Publ. Math. IHES 51 (1980), 137-173.
  • [K2] A. Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems 2 (1982), 339-367.
  • [KH] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, New York, 1995.
  • [KM] A. Katok and L. Mendoza, Dynamical systems with non-uniformly hyperbolic structure, supplement to [KH], 659-700.
  • [Kn] G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds, preprint, 1996.
  • [KT] T. Krüger and S. Troubetzkoy, Markov partitions and shadowing for diffeomorphisms with no zero exponents, preprint, 1997.
  • [LY] T.-Y. Li and J. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc. 235 (1978), 183-192.
  • [Li] D. Lind, Perturbations of shifts of finite type, SIAM J. Discrete Math. 2 (1989), 350-365.
  • [MlT] J. Milnor and W. Thurston, On iterated maps of the interval, in: Dynamical Systems (College Park, Md., 1986-87), Lecture Notes in Math. 1342, Springer, Berlin, 1988, 465-563.
  • [M1] M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 27 (1979), 167-169.
  • [M2] M. Misiurewicz, Continuity of entropy revisited, in: Dynamical Systems and Applications, World Sci. Ser. Appl. Anal. 4, World Sci., 1995, 495-503.
  • [MS1] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Astérisque 50 (1977), 299-310.
  • [MS2] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math. 67 (1980), 45-63.
  • [P] Ya. Pesin, Characteristic exponents and smooth ergodic theory, Russian Math. Surveys 32 (1977), 55-114.
  • [Pu] C. C. Pugh, The $C^1 + α$ hypothesis in Pesin theory, Publ. Math. IHES 59 (1984), 143-161.
  • [Q] A. Quas, Invariant densities for $C^1$ maps, Studia Math. 120 (1996), 83-88.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-fmv157i2p245bwm
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