Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Nielsen-Thurston reducibility
and renormalization


Authors: Olivier Courcelle, Jean-Marc Gambaudo and Charles Tresser
Journal: Proc. Amer. Math. Soc. 125 (1997), 3051-3058
MSC (1991): Primary 58F99
DOI: https://doi.org/10.1090/S0002-9939-97-04159-2
MathSciNet review: 1425117
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Consider an orientation preserving homeomorphism $f$ of the 2-disk with an infinite set of nested periodic orbits $\{\mathcal {O}_n\}_{n\ge 1}$, such that, for all $m>1$, the restriction of $f$ to the complement of the first $m$ orbits, from $\mathcal {O}_1$ to $\mathcal {O}_m$, is $m-1$ times reducible in the sense of Nielsen and Thurston. We define combinatorial renormalization operators for such maps, and study the fixed points of these operators. We also recall the corresponding theory for endomorphisms of the interval, and give elements of comparison of the theories in one and two dimensions.


References [Enhancements On Off] (What's this?)

  • [BORT] H. Bass, M. V. Otero-Espinar, D. Rockmore, and C. Tresser, Cyclic Renormalization and Automorphism Groups of Rooted Trees, Lecture Notes in Mathematics 1621 (Springer, Berlin, 1996). CMP 96:13
  • [BF] R. Bowen and J. Franks, The periodic points of maps of the disk and the interval, Topology 15 (1976), 337-342. MR 55:4283
  • [CT] P. Coullet and C. Tresser, Itérations d'endomorphismes et groupe de renormalisation, J. Phys. C5 (1978), 25-28.
  • [GGH] J. M. Gambaudo, J. Guaschi, and T. Hall, Period-multiplying cascades for diffeomorphisms of the disk, Math. Proc. Cambridge Philos. Soc. 116 (1994), 359-374. MR 95e:58129
  • [GStT] J. M. Gambaudo, S. van Strien, and C. Tresser, There exists a $C^\infty $ Kupka-Smale diffeomorphism on $S^2$ with neither sinks nor sources, Nonlinearity 2 (1989), 287-304. MR 90b:58154
  • [GSuT] J. M. Gambaudo, D. Sullivan, and C. Tresser, Infinite cascades of braids and smooth dynamics, Topology 33 (1994), 85-94. MR 95a:58078
  • [GT] J. M. Gambaudo and C. Tresser, Self-similar constructions in smooth dynamics. Rigidity, smoothness and dimension, Comm. Math. Phys. 150 (1992), 45-58. MR 93j:58084
  • [Fe] M. J. Feigenbaum, Quantitative universality for a class of non-linear transformations, J. Stat. Phys. 19 (1978), 25-52. MR 58:18601
  • [FY] J. Franks and L. S. Young, A $C^2$ Kupka-Smale diffeomorphism of the disk with no sources or sinks, in Dynamical Systems and Turbulence (Warwick 1980), Lecture Notes in Mathematics 898 (Springer-Verlag, Berlin, 1981). MR 83j:58095
  • [Ni] J. Nielsen, ``Investigations in the topology of closed orientable surfaces I, II, and III,'' Translation by John Stillwell of: ``Untersuchungen zur Topologie der geschlossenen zweiseitingen Flaïchen.'' In Jakob Nielsen: Collected Mathematical Papers (Birkhäuser, Boston, 1986). MR 88a:01070a; MR 88a:01070b
  • [Th] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417-431. MR 89k:57023
  • [TC] C. Tresser and P. Coullet, Itérations d'endomorphismes et groupe de renormalisation, C. R. Acad. Sci. Paris Sér. A 287 (1978), 577-580. MR 80b:58043

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 58F99

Retrieve articles in all journals with MSC (1991): 58F99


Additional Information

Olivier Courcelle
Affiliation: Section de Mathématiques, Université de Genève, CP240, CH1211 Genève 24, Suisse
Email: courcell@divsun.unige.ch

Jean-Marc Gambaudo
Affiliation: INLN, 1361 route des lucioles, Sophia-Antipolis, 06560 Valbonne, France
Email: jmga@ecu.unice.fr

Charles Tresser
Affiliation: IBM, P.O. Box 218, Yorktown Heights, New York 10598
Email: tresser@watson.ibm.com

DOI: https://doi.org/10.1090/S0002-9939-97-04159-2
Received by editor(s): October 24, 1995
Communicated by: Linda Keen
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society