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

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 fa, b : (x, y)↦ (x²-a-by, x) defined on R² 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≠ 0 (including the case $b < 0$ as well). The method of the proof is first to regard fa, b as a complex dynamical system in C² and second to introduce the new Markov-like partition in R² constructed by us [On parameter loci of the Hénon family. Comm. Math. Phys. 361 (2) (2018), 343–414].