Hostname: page-component-669899f699-ggqkh Total loading time: 0 Render date: 2025-04-29T16:41:32.477Z Has data issue: false hasContentIssue false
  • English
  • Français

Symbolic dynamics for Hénon maps near the boundary of the horseshoe locus

Published online by Cambridge University Press:  23 May 2024

YUKI HIRONAKA  and
YUTAKA ISHII
YUKI HIRONAKA
Affiliation:
Department of Mathematics, Kyushu University, Motooka, Fukuoka 819-0395, Japan (e-mail: [email protected])
YUTAKA ISHII*
Affiliation:
Department of Mathematics, Kyushu University, Motooka, Fukuoka 819-0395, Japan (e-mail: [email protected])
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 1 , January 2025 , pp. 140 - 174
Check for updates
Copyright
© The Author(s), 2024. 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.)

Article purchase

Temporarily unavailable

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
Bedford, 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
Bedford, E. and Smillie, J.. Polynomial diffeomorphisms of ${\mathbf{C}}^2$ . VIII. Quasi-expansion. Amer. J. Math. 124(2) (2002), 221–271.CrossRefGoogle Scholar
Bedford, 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
Ishii, Y. and Smillie, J.. Homotopy shadowing. Amer. J. Math. 132(4) (2010), 987–1029.CrossRefGoogle Scholar