No CrossRef data available.
Article contents
Symbolic dynamics for Hénon maps near the boundary of the horseshoe locus
Published online by Cambridge University Press: 23 May 2024
Abstract
Bedford and Smillie [A symbolic characterization of the horseshoe locus in the Hénon family. Ergod. Th. & Dynam. Sys. 37(5) (2017), 1389–1412] classified the dynamics of the Hénon map 
$f_{a, b} : (x, y)\mapsto (x^2-a-by, x)$ defined on 
$\mathbb {R}^2$ in terms of a symbolic dynamics when 
$(a, b)$ is close to the boundary of the horseshoe locus. The purpose of the current article is to generalize their results for all 
$b\ne 0$ (including the case 
$b < 0$ as well). The method of the proof is first to regard 
$f_{a, b}$ as a complex dynamical system in 
$\mathbb {C}^2$ and second to introduce the new Markov-like partition in 
$\mathbb {R}^2$ constructed by us [On parameter loci of the Hénon family. Comm. Math. Phys. 361(2) (2018), 343–414].
- Type
- Original Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press
References
Arai, Z.. On hyperbolic plateaus of the Hénon map. Exp. Math. 16(2) (2007), 181–188.CrossRefGoogle Scholar
Arai, Z.. All programs for computer-assisted proofs and their instructions are available at: http://www.is.c.titech.ac.jp/zin/hironaka-ishii/.Google Scholar
Arai, Z. and Ishii, Y.. On parameter loci of the Hénon family. Comm. Math. Phys. 361(2) (2018), 343–414.CrossRefGoogle Scholar
Bedford, E. and Dujardin, R.. Topological and geometric hyperbolicity criteria for polynomial automorphisms of
${\mathbb{C}}^2$
. Ergod. Th. & Dynam. Sys. 42(7) (2022), 2151–2171.CrossRefGoogle Scholar
${\mathbb{C}}^2$
. Ergod. Th. & Dynam. Sys. 42(7) (2022), 2151–2171.CrossRefGoogle ScholarBedford, E., Lyubich, M. and Smillie, J.. Polynomial diffeomorphisms of
${\mathbf{C}}^2$
: IV. The measure of maximal entropy and laminar currents. Invent. Math. 112(1) (1993), 77–125.CrossRefGoogle Scholar
${\mathbf{C}}^2$
: IV. The measure of maximal entropy and laminar currents. Invent. Math. 112(1) (1993), 77–125.CrossRefGoogle ScholarBedford, E. and Smillie, J.. Polynomial diffeomorphisms of
${\mathbf{C}}^2$
. VIII. Quasi-expansion. Amer. J. Math. 124(2) (2002), 221–271.CrossRefGoogle Scholar
${\mathbf{C}}^2$
. VIII. Quasi-expansion. Amer. J. Math. 124(2) (2002), 221–271.CrossRefGoogle ScholarBedford, E. and Smillie, J.. Real polynomial diffeomorphisms with maximal entropy: Tangencies. Ann. of Math. (2) 160(1) (2004), 1–26.CrossRefGoogle Scholar
Bedford, E. and Smillie, J.. Real polynomial diffeomorphisms with maximal entropy. II. Small Jacobian. Ergod. Th. & Dynam. Sys. 26(5) (2006), 1259–1283.CrossRefGoogle Scholar
Bedford, E. and Smillie, J.. A symbolic characterization of the horseshoe locus in the Hénon family. Ergod. Th. & Dynam. Sys. 37(5) (2017), 1389–1412.CrossRefGoogle Scholar
Ishii, Y.. Hyperbolic polynomial diffeomorphisms of
${\mathbb{C}}^2$
. I: A non-planar map. Adv. Math. 218(2) (2008), 417–464.CrossRefGoogle Scholar
${\mathbb{C}}^2$
. I: A non-planar map. Adv. Math. 218(2) (2008), 417–464.CrossRefGoogle ScholarIshii, Y. and Smillie, J.. Homotopy shadowing. Amer. J. Math. 132(4) (2010), 987–1029.CrossRefGoogle Scholar