Infinitely renormalizable diffeomorphisms of the disk at the boundary of chaos

Catsigeras, Eleonora; Gambaudo, Jean-Marc; Moreira, Fernando

doi:10.1090/S0002-9939-98-03945-8

Infinitely renormalizable diffeomorphisms of the disk at the boundary of chaos
HTML articles powered by AMS MathViewer

by Eleonora Catsigeras, Jean-Marc Gambaudo and Fernando Moreira PDF

Proc. Amer. Math. Soc. 126 (1998), 297-304 Request permission

Abstract:

We show that, among area contracting embeddings of the 2-disk, infinitely renormalizable maps with a bounded geometry either have positive topological entropy or correspond to a cascade of period doubling.

References

Rufus Bowen and John Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), no. 4, 337–342. MR 431282, DOI 10.1016/0040-9383(76)90026-4
E. Catsigeras, “Cascades of period doubling of stable codimension one,” IMPA’s thesis, 1994.
P. Coullet and C. Tresser, “Itérations d’endomorphismes et groupe de renormalisation,” J. Phys. C5 (1978),25–28.
J.-M. Gambaudo, S. van Strien, and C. Tresser, Hénon-like maps with strange attractors: there exist $C^\infty$ Kupka-Smale diffeomorphisms on $S^2$ with neither sinks nor sources, Nonlinearity 2 (1989), no. 2, 287–304. MR 994094, DOI 10.1088/0951-7715/2/2/005
Jean-Marc Gambaudo and Charles Tresser, How horseshoes are created, Instabilities and nonequilibrium structures, III (Valparaíso, 1989) Math. Appl., vol. 64, Kluwer Acad. Publ., Dordrecht, 1991, pp. 13–25. MR 1177837, DOI 10.1007/978-94-011-3442-2_{2}
Jean-Marc Gambaudo and Charles Tresser, Self-similar constructions in smooth dynamics: rigidity, smoothness and dimension, Comm. Math. Phys. 150 (1992), no. 1, 45–58. MR 1188495, DOI 10.1007/BF02096564
Mitchell J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), no. 1, 25–52. MR 501179, DOI 10.1007/BF01020332
J. Hu, “Renormalization, rigidity and universality in bifurcation theory”, Ph. D. Dissertation, CUNY (1995).
J. Hu and C. Tresser, “Period Doubling, Entropy, and Renormalization” preprint (1995).
Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
MichałMisiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 2, 167–169 (English, with Russian summary). MR 542778
Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622
Charles Tresser and Pierre Coullet, Itérations d’endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 7, A577–A580 (French, with English summary). MR 512110