Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-22T19:05:22.288Z Has data issue: false hasContentIssue false
  • English
  • Français

On the finiteness of attractors for piecewise$C^{2}$ maps of the interval

Published online by Cambridge University Press:  04 December 2017

P. BRANDÃO ,
J. PALIS  and
V. PINHEIRO
P. BRANDÃO
Affiliation:
Impa, Estrada Dona Castorina 110, Rio de Janeiro, Brazil email paulo@impa.br, jpalis@impa.br
J. PALIS
Affiliation:
Impa, Estrada Dona Castorina 110, Rio de Janeiro, Brazil email paulo@impa.br, jpalis@impa.br
V. PINHEIRO
Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil email viltonj@ufba.br
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider piecewise $C^{2}$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^{2}$ non-flat map of the interval displays only a finite number of non-periodic attractors.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 7 , July 2019 , pp. 1784 - 1804
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avila, A., Lyubich, M. and de Melo, W.. Regular or stochastic dynamics in real analytic families of unimodal maps. Invent. Math. 154 (2003), 451–550.Google Scholar
Blokh, A. M. and Lyubich, M.. Nonexistence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. II. The smooth case. Ergod. Th. & Dynam. Sys. 9(4) (1989), 751–758.Google Scholar
Blokh, A. and Lyubich, M.. Measurable dynamics of S-unimodal maps of the interval. Ann. Sci. Éc. Norm. Supér. 24 (1991), 545–573.Google Scholar
Brandão, P., Palis, J. and Pinheiro, V.. On the finiteness of attractors for one-dimensional maps with discontinuities. Preprint, 2013, arXiv:1401.0232.Google Scholar
Bruin, H., Shen, W. and van Strien, S.. Existence of unique SRB-measures is typical for unimodal families. Ann. Sci. Éc. Norm. Supér. 39 (2006), 381–414.Google Scholar
Guckenheimer, J. and Williams, R. F.. Structural stability of Lorenz attractors. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 59–72.Google Scholar
Lyubich, M.. Ergodic theory for smooth one dimensional dynamical systems, Preprint, arXiv:9201286, 1991.Google Scholar
Lyubich, M.. Almost every real quadratic map is either regular or stochastic. Ann. of Math. (2) 156(1) (2002), 1–78.Google Scholar
Mañé, R.. Hyperbolicity, sinks and measure in one-dimensional dynamics. Commun. Math. Phys. 100 (1985), 495–524 . Commun. Math. Phys. 112 (1987), 721–724 (erratum).Google Scholar
Martens, M., de Melo, W. and van Strien, S.. Julia–Fatou–Sullivan theory for real one- dimensional dynamics. Acta Math. 168 (1992), 273–318.Google Scholar
de Melo, W. and van Strien, S.. A structure theorem in one dimensional dynamics. Ann. of Math. (2) 129 (1989), 519–546.Google Scholar
de Melo, W. and van Strien, S.. One Dimensional Dynamics. Springer, Berlin, 1993.Google Scholar
Palis, J.. A global view of dynamics and a conjecture on the denseness of finitude of attractors. Astérisque 261 (2000), 339–351.Google Scholar
Palis, J.. A global perspective for non-conservative dynamics. Ann. Inst. H. Poincaré. Anal. Non Linéaire 22 (2005), 487–507.Google Scholar
Vilton, P.. Expanding measures. Ann. Inst. H. Poincaré. Anal. Non Linéaire 28 (2011), 889–939.Google Scholar
van Strien, S. and Vargas, E.. Real bounds, ergodicity and negative Schwarzian for multimodal maps. J. Amer. Math. Soc. 17 (2004), 749–782.Google Scholar