Abstract
Consider the two-parameter family of real analytic maps\(F_{a,b} :x \mapsto x + a + \tfrac{b}{{2\pi }}\) sin(2πx) which are lifts of degree one endomorphisms of the circle. The purpose of this paper is to provide a proof that for any closed intervalI, the set of mapsF a,b whose rotation interval isI, form a contractible set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alsedà, L., Mañosas, F.: Kneading theory and rotation intervals for a class of circle maps of degree one. Nonlinearity3, 413–452 (1990)
Arnold, V.I.: Small denominators, I. Mappings of the circumference onto itself. AMS Trans.46, 213–284 (1965)
Block, L., Franke, J.: Existence of periodic points for maps ofS 1,. Invent. Math.22, 69–73 (1973)
Boyland, P.L.: Bifurcations of circle maps: Arnol'd tongues bistability and rotation intervals. Commun. Math. Phys.106, 353–381 (1986)
Chenciner, A., Gambaudo, J.M., Tresser, C.: Une remarque sur la structure des endomorphismes de degré un du cercle. C.R. Acad. Sc. Paris série I,299, 771–773 (1984)
Dawson, S., Galeeva, R., Milnor, J., Tresser, C.: A monotonicity conjecture for real cubic maps. IMS-SUNY Stonybrook preprint series, #1993/11
Denjoy, A.: Sur les courbes définies par les équations différentielles à la surface du tore. J. de Math. Pures et Appl.11, 333–375 (1932)
Epstein, A.: Towers of finite type complex analytic maps. Ph.D. Thesis, CUNY, (1993)
Fatou, P.: Sur les equations fonctionnelles. Bull. Soc. Math. France,47, 161–271 (1919)
Fatou, P.: Sur les equations fonctionnelles. Bull. Soc. Math. France,48, 33–94, 208–314 (1920)
Galeeva, R., Martens, M., Tresser, C.: Inducing, slopes, and conjugacies. IMS-SUNY Stonybrook preprint series, #1994/4
Gardiner, F.: Teichmüller Theory and Quadratic Differentials. New York: Wiley-Interscience, 1987
Goldberg, L.R., Keen, L.: A finiteness theorem for a dynamical class of entire functions. J. Ergodic Th. and Dyn. Sys.6, 183–192 (1986)
Herman, M.R.: Sur la conjugaison differentiable des diffémorphismes à des rotations. Pub. Math. I.H.E.S.49, 5–234 (1980)
Keen, L.: Dynamics of holomorphic self-maps ofC *. In: D. Drasin et al., editor, Holomorphic Functions and Moduli. M.S.R.I., Springer, 1988
Keen, L.: Julia sets. In: Keen, L., Devaney, R. editors, Chaos and Fractals: The Mathematics Behind the Computer Graphics, chapter Julia Sets, pages 57–74. Providence, KI, AMS, 1989
Kotus, J.: Iterated holomorphic maps on the punctured plane. Preprint #358, Polish Acad. of Math., 1986
Lehto, O.: Univalent Functions and Teichmüller Spaces. Berlin-Heidelberg-New York: Springer, 1987
Lyubich, M.: Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems. I. The case of negative Schwarzian derivative. Erg. Th. & Dyn. Syst.9, 737–750 (1989)
Mackay, R.S., Tresser, C.: Transition to topological chaos for circle maps. Physica19D, 206–237 (1986) &29D, 427 (1988)
Martens, M., de Melo, W., van Strien, S.: Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math.168, 273–318 (1992)
McMullen, C., Sullivan, D.P.: Quasiconformal homeomorphisms and dynamics: III. Preprint
McMullen, C.T.: Families of rational maps and iterative root-finding algorithms. Ann. Math.125, 467–493 (1987)
de Melo, W., van Strien, S.: One-Dimensional Dynamics. Berlin, Heidelberg, New York: Springer, 1993
Milnor, J.: Dynamics in one complex variable: Introductory lectures. IMS-SUNY Stonybrook preprint series, #1990/5
Milnor, J., Thurston, W.: On iterated maps of the interval. Springer Lecture Notes,1342, 465–563 (1988)
Misiurewicz, M.: Twist sets for maps of the circle. Erg. Th. & Dyn. Syst.4, 391–404 (1984)
Newhouse, S., Palis, J., Takens, F.: Stable families of diffeomorphisms. Publ. Math. I.H.E.S.57, 5–71 (1983)
Poincaré, H.: Sur les courbes définies par des équations différentielles. J. Math. Pures et Appl., 4eme série,1, 167–244 (1885 Also in: O Euvres Complétes, t 1 Paris: Gauthier-Villars, 1951
Siegel, C.L.: Bermerkg zu einem Satz von Jakob Nielsen. In: Gesammelte Abhandlungen, Vol.3, Berlin-Heidelberg-New York: Springer-Verlag, 1966, pp. 92–96
Sullivan, D.P.: Quasiconformal homeomorphisms and dynamics: I. Ann. Math.122, 401–418 (1985)
Uherka, D.J., Tresser, C., Galeeva, R., Campbell, D.K.: Solvable models for the quasi-periodic transition to chaos. Phys. Let. A170, 189–194 (1992), erratum, A177, 461 (1993), and to appear
Author information
Authors and Affiliations
Additional information
Communicated by Ya. G. Sinai
Supported in part by NSF GRANT DMS-9205433, Inst. Math. Sciences, SUNY-StonyBrook and I.B.M.
Rights and permissions
About this article
Cite this article
Epstein, A., Keen, L. & Tresser, C. The set of maps\(F_{a,b} :x \mapsto x + a + \tfrac{b}{{2\pi }}\) sin(2πx) with any given rotation interval is contractiblewith any given rotation interval is contractible. Commun.Math. Phys. 173, 313–333 (1995). https://doi.org/10.1007/BF02101236
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02101236