Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T08:24:31.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Kneading sequences for toy models of Hénon maps

Part of: Low-dimensional dynamical systems Topological dynamics

Published online by Cambridge University Press:  13 November 2020

ERMERSON ARAUJO Open the ORCID record for ERMERSON ARAUJO [Opens in a new window]
ERMERSON ARAUJO*
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, Centro de Ciências, Campus do Pici, Fortaleza – CE, CEP 60440-900, Brasil
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of this article is to study the relation between combinatorial equivalence and topological conjugacy, specifically how a certain type of combinatorial equivalence implies topological conjugacy. We introduce the concept of kneading sequences for a setting that is more general than one-dimensional dynamics: for the two-dimensional toy model family of Hénon maps introduced by Benedicks and Carleson, we define kneading sequences for their critical lines, and prove that these sequences are a complete invariant for a natural conjugacy class among the toy model family. We also establish a version of Singer’s theorem for the toy model family.

Keywords

combinatorial equivalencekneading sequencesunimodal maps

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3521 - 3540
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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

Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. of Math. (2) 133(1) (1991), 73–169.CrossRefGoogle Scholar
de Carvalho, A. and Hall, T.. How to prune a horseshoe. Nonlinearity 15(3) (2002), R19–R68.10.1088/0951-7715/15/3/201CrossRefGoogle Scholar
de Carvalho, A. and Hall, T.. Symbolic dynamics and topological models in dimensions 1 and 2. Topics in Dynamics and Ergodic Theory (London Mathematical Society Lecture Note Series, 310). Cambridge University Press, Cambridge, 2003, pp. 40–59.CrossRefGoogle Scholar
Crovisier, S. and Pujals, E.R.. Strongly dissipative surface diffeomorphisms. Comment. Math. Helv. 93(2) (2018), 377–400.CrossRefGoogle Scholar
Cvitanović, P.. Periodic orbits as the skeleton of classical and quantum chaos. Phys. D 51(1–3) (1991), 138–151.CrossRefGoogle Scholar
Cvitanović, P., Gunaratne, G.H. and Procaccia, I.. Topological and metric properties of Hénon-type strange attractors. Phys. Rev. A (3) 38(3) (1988), 1503–1520.CrossRefGoogle ScholarPubMed
Ishii, Y.. Towards a kneading theory for Lozi mappings. I. A solution of the pruning front conjecture and the first tangency problem. Nonlinearity 10(3) (1997), 731–747.CrossRefGoogle Scholar
Matheus, C., Moreira, C.G. and Pujals, E. R.. Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models. Ann. Sci. Éc. Norm. Supér. (4) 46(6) (2013), 857–878.CrossRefGoogle Scholar
de Melo, W. and van Strien, S.. One-Dimensional Dynamics ( Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25). Springer, Berlin, 1993.CrossRefGoogle Scholar
Mendoza, V.. Proof of the pruning front conjecture for certain Hénon parameters. Nonlinearity 26(3) (2013), 679–690.CrossRefGoogle Scholar
Milnor, J. and Thurston, W.. On iterated maps of the interval. Dynamical Systems (College Park, MD, 1986) (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 465–563.Google Scholar
Rand, D.. The topological classification of Lorenz attractors. Math. Proc. Cambridge Philos. Soc. 83(3) (1978), 451–460.CrossRefGoogle Scholar