More on the Concept of Anti-integrability for H\'enon Maps

For the family of H\'{e}non maps (x,y)↦ (√a(1-x²)-b y,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 H\'{e}non 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=r̂ for any r̂>0 provided that the one dimensional quadratic map x↦ 1r̂(1-x²) is hyperbolic.