Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2000 | 164 | 2 | 115-141
Tytuł artykułu

Dynamics on Hubbard trees

Treść / Zawartość
Warianty tytułu
Języki publikacji
It is well known that the Hubbard tree of a postcritically finite complex polynomial contains all the combinatorial information on the polynomial. In fact, an abstract Hubbard tree as defined in [23] uniquely determines the polynomial up to affine conjugation. In this paper we give necessary and sufficient conditions enabling one to deduce directly from the restriction of a quadratic Misiurewicz polynomial to its Hubbard tree whether the polynomial is renormalizable, and in this case, of which type. Moreover, we study dynamical features such as entropy, transitivity or periodic structure of the polynomial restricted to the Hubbard tree, and compare them with the properties of the polynomial on its Julia set. In other words, we want to study how much of the "dynamical information" about the polynomial is captured by the Hubbard tree.
Opis fizyczny
  • Departament de Matemàtiques, Facultat de Ciències (Edifici Cc), Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
  • Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
  • [1] R. Adler, A. Konheim and J. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319.
  • [2] Ll. Alsedà, S. Baldwin, J. Llibre and M. Misiurewicz, Entropy of transitive tree maps, Topology 36 (1996), 519-532.
  • [3] Ll. Alsedà, S. Kolyada, J. Llibre and Ľ. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc. 351 (1999), 1551-1573.
  • [4] Ll. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Adv. Ser. Nonlinear Dynamics 5, World Sci., Singapore, 1993.
  • [5] Ll. Alsedà, M. A. del Río and J. A. Rodríguez, A splitting theorem for transitive maps, J. Math. Anal. Appl. 232 (1999), 359-375.
  • [6] Ll. Alsedà, M. A. del Río and J. A. Rodríguez, Cofiniteness of the set of periods for totally transitive tree maps, Internat. J. Bifur. Chaos 9 (1999), 1877-1880.
  • [7] A. Beardon, Iteration of Rational Functions, Grad. Texts in Math. 132, Springer, New York, 1991.
  • [8] P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. 11 (1984), 85-141.
  • [9] L. Block, J. Guckenheimer, M. Misiurewicz and L.-S. Young, Periodic points and topological entropy of one dimensional maps, in: Global Theory of Dynamical Systems, Lecture Notes in Math. 819, Springer, Berlin, 1980, 18-34.
  • [10] A. M. Blokh, On transitive mappings of one-dimensional branched manifolds, in: Diff.-Difference Equations and Problems of Mathematical Physics, Inst. of Math., Kiev, 1984, 3-9 (in Russian).
  • [11] A. M. Blokh, On the connection between entropy and transitivity for one-dimensional mappings, Russian Math. Surveys 42 (1987), 165-166.
  • [12] L. Carleson and T. Gamelin, Complex Dynamics, Springer, New York, 1993.
  • [13] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Math. 527, Springer, Berlin, 1976.
  • [14] A. Douady et J. Hubbard, Etude dynamique des polynômes complexes, part I, Publ. Math. Orsay, 1984-1985.
  • [15] A. Douady et J. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Ecole Norm. Sup. 18 (1985), 287-343.
  • [16] A. Eremenko and M. Lyubich, The dynamics of analytic transformations, Leningrad Math. J. 1 (1990), 563-634.
  • [17] F. R. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York, 1959.
  • [18] J. Hubbard, Puzzles and quadratic tableaux (according to Yoccoz), preprint, 1990.
  • [19] C. T. McMullen, Complex Dynamics and Renormalization, Princeton Univ. Press, 1994.
  • [20] J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, Vieweg, 1999.
  • [21] J. Milnor, Local connectivity of Julia sets: expository lectures, Stony Brook preprint no. 1990/5 (1992).
  • [22] C. L. Petersen, On the Pommerenke-Levin-Yoccoz inequality, Ergodic Theory Dynam. Systems 13 (1993), 785-806.
  • [23] A. Poirier, On postcritically finite polynomials. Part two: Hubbard trees, Stony Brook preprint no. 1993/7.
  • [24] Se E. Seneta, Non-Negative Matrices and Markov Chains, Springer Ser. in Statist., Springer, Berlin, 1981.
  • [25] N. Steinmetz, Rational Iteration: Complex Analytic Dynamical Systems, de Gruyter, 1993.
  • [26] P. Walters, An Introduction to Ergodic Theory, Grad. Texts in Math. 79, Springer, New York, 1982.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.