Dynamics of transcendental Hénon maps

Abstract

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental Communicated by Ngaiming Mok.

Nonexistence of wandering domains for strongly dissipative infinitely renormalizable Hénon maps at the boundary of chaos

“…The parabola is the graph of a unimodal map. The points p(0), p (1) , and p (2) are defined in Definition 3.1 (1) p (2) The class of analytic unimodal maps f : I → I is denoted by U. For a unimodal map f ∈ U, let c (0) = c (0) ( f ) ∈ I be the critical point of f .…”

“…Definition 3.1 Assume that f ∈ U has a unique fixed point p(0) ∈ I with a negative multiplier. Let p (1) = p(0) and p (2) be the point such that f ( p (2) ) = p (1) and p (2) > c (0) . Also, let A = (−1, p (1) ) ∪ ( p (2) , 1), B = ( p (1) , p(0)), and C = ( p(0), p (2) ).…”

“…Here, the theorem is restated in our coordinate system. (1) , c (1) ]. (1) ) = − 1 λ c (1) and g (c (1) ) = −λ.…”

“…The reader can refer to the survey [62] for more details about other relevant works. For complex Hénon maps, 2 a recent paper by Arosio et al [1] found a transcendental Hénon map exhibiting a wandering domain. Nevertheless, the problem is still unsolved [4] for complex polynomial Hénon maps.…”

“…The maps are a generalization of classical Hénon maps [36] (a two-parameter family of polynomial maps) to the analytic settings and an extension of unimodal maps to higher dimensions. For renormalization purposes, the maps under consideration are all real analytic and strongly dissipative (the Jacobian | ∂ ∂ y | is small 1 ). Strongly dissipative Hénon-like maps are the maps that are closest to unimodal maps, with which they share many dynamical properties.…”