More on the concept of anti-integrability for Hénon maps

For the family of Hénon maps (x,y)↦(a(1−x2)−by,x) of R2, the so-called anti-integrable (AI) limit concerns the limit a → ∞ with fixed Jacobian b. At the AI limit, the dynamics reduces to a subshift of finite type. There is a one-to-one correspondence between sequences allowed by the subshift and the AI orbits. The theory of anti-integrability says that each AI orbit can be continued to becoming a genuine orbit of the Hénon map for a sufficiently large (and fixed Jacobian). In this paper, we assume b is a smooth function of a and show that the theory can be extended to investigating the limit lima→∞b/a=ν for any ν > 0 provided that the one dimensional quadratic map x↦1ν(1−x2) is hyperbolic.