×

zbMATH — the first resource for mathematics

Julia-Fatou-Sullivan theory for real one-dimensional dynamics. (English) Zbl 0761.58007
Denoting by \(N\) either the circle \(S'\) or a compact interval of the real line, the authors study the dynamics of a smooth endomorphism \(f\) of \(N\) which is not a homeomorphism (if \(f\) is a homeomorphism then it corresponds to a degree \(\pm1\) rational map of the Riemann sphere). The singular set of \(f\), \(\text{Sing}(f)\), is defined as the union of the set of turning points of \(f\) (a turning point is a critical noninflection point) and the boundary points of \(N\) — in the case when \(f\) has turning points — and as the set of fixed points of \(f\) — when \(f\) has no turning points. The Julia set of \(f\), \(J(f)\), is defined to be the limit set of \(\text{Sing}(f)\) and it is forward invariant. Thus its complement, called Fatou set \(F(f)\), is backward invariant \((f^{- 1}(F(f))\subset F(f))\).
The main result of the paper is the following: If \(f:N\to N\) is a smooth map such that all its critical points are non-flat (i.e., some higher derivative is non-zero), then all the connected components of \(F(f)\) are eventually periodic (i.e., eventually mapped into a periodic component of \(F(f)\)) and the number of periodic components of \(F(f)\) is finite.

MSC:
37E99 Low-dimensional dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] [BL]Blokh, A. M. &Lyubich, M. Ju., Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems 2. The smooth case.Ergodic Theory Dynamical Systems, 9 (1989), 751–758. · Zbl 0665.58024
[2] [CE]Collet, P. & Eckmann, J.,Iterated Maps of the Interval as Dynamical Systems. Birkhäuser, 1980. · Zbl 0458.58002
[3] [D]Denjoy, A., Sur les courbes definie par les equations differentielles a la surface du tore.J. Math. Pures Appl. 9, 11 (1932), 333–375. · JFM 58.1124.04
[4] [Fa]Fatou, P., Sur les equations fonctionnelles.Bull. Soc. Math. France, 47 (1919), 161–271; 48 (1920), 33–94; 48 (1920), 208–314. · JFM 47.0921.02
[5] [Gu1]Guckenheimer, J., Sensitive dependence on initial conditions for one dimensional maps.Comm. Math. Phys., 70 (1979), 133–160. · Zbl 0429.58012 · doi:10.1007/BF01982351
[6] [Gu2]–, Limit sets ofS-unimodal maps with zero entropy.Comm. Math. Phys., 110 (1987), 655–659. · Zbl 0625.58027 · doi:10.1007/BF01205554
[7] [Ha]Hall, C. R., AC Denjoy counterexample.Ergodic Theory Dynamical Systems, 1 (1981), 261–272.
[8] [J]Julia, G., Memoire sur l’iteration des fonctions rationnelles.J. de Math., 8 (1918), 47–245. · JFM 46.0520.06
[9] [L]Lyubich, M. Ju., Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems 1. The case of negative Schwarzian derivative.Ergodic Theory Dynamical Systems, 9 (1989), 737–750. · Zbl 0665.58023
[10] [Ma]Mañé, R., Hyperbolicity, sinks and measure in one dimensional dynamics.Comm. Math. Phys., 100 (1985), 495–524; Erratum.Comm. Math. Phys., 112 (1987), 721–724. · Zbl 0583.58016 · doi:10.1007/BF01217727
[11] [Me]de Melo, W., A finiteness problem for one dimensional maps.Proc. Amer. Math. Soc., 101 (1987), 721–727. · Zbl 0656.54029
[12] [Mi]Misiurewicz, M., Absolutely continuous measure for certain maps of an interval.Publ. Math. I.H.E.S., 53 (1981), 17–51. · Zbl 0477.58020
[13] [MS1]de Melo, W. &van Strien, S. J., A structure theorem in one-dimensional dynamics.Ann. of Math., 129 (1989), 519–546. · Zbl 0737.58020 · doi:10.2307/1971516
[14] [MS2] –, Schwarzian derivative and beyond.Bull. Amer. Math. Soc., 18 (1988), 159–162. · Zbl 0651.58019 · doi:10.1090/S0273-0979-1988-15633-9
[15] [MMMS]Martens, M., de Melo, W., Mendes, P. &van Strien, S., Cherry flows on the torus: towards a classification.Ergodic Theory Dynamical Systems, 10 (1990), 531–554. · Zbl 0694.58025
[16] [MT]Milnor, J. & Thurston, W., On iterated maps of the interval: I, II. Preprint Princeton, 1977.
[17] [Si]Singer, D., Stable orbits and bifurcations of maps of the interval.SIAM J. Appl. Math., 35 (1978), 260–267. · Zbl 0391.58014 · doi:10.1137/0135020
[18] [Sc]Schwartz, A., A generalization of a Poincaré-Bendixon theorem to closed two dimensional manifolds.Amer. J. Math., 85 (1963), 453–458. · Zbl 0116.06803 · doi:10.2307/2373135
[19] [SI]Sharkovskii, A. N. &Ivanov, A. F.,C mappings of an interval with attracting cycles of arbitrary large periods.Ukrain. Mat. Zh., 4 (1983), 537–539.
[20] [Str1]van Strien, S., On the bifurcations creating horseshoes, inDynamical Systems and Turbulence, Warwick, 1980. Lecture Notes in Mathematics, 898 (1981), 316–351. Springer-Verlag, Berlin.
[21] [Str2] –, Smooth dynamics on the interval, inNew Direction in Chaos. Cambridge University Press, Cambridge, 1987, pp. 57–119.
[22] [Str3] –, Hyperbolicity and invariant measures for generalC 2 interval maps satisfying the Misiurewicz condition.Comm. Math. Phys., 128 (1990), 437–496. · Zbl 0702.58020 · doi:10.1007/BF02096868
[23] [Su]Sullivan, D., Quasiconformal homeomorphisms and dynamics I: a solution of Fatou-Julia problem on wandering domains.Ann. of Math., 122 (1985), 401–418; II: Structural stability implies hyperbolicity for Kleinian groups.Acta Math., 155 (1985), 243–260. · Zbl 0589.30022 · doi:10.2307/1971308
[24] [Y]Yoccoz, J. C., Il n y a pas de contre-examples de Denjoy analytique.C. R. Acad. Sci. Paris Sér. I Math., 298 (1984), 141–144. · Zbl 0573.58023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.