Generalized Hénon maps: The cubic diffeomorphisms of the plane
The paper is devoted to the study of the dynamics of generalized Hénon maps \(h:\mathbb R^2\to \mathbb R^2\), \[ h(x,y)=(y+p(x),-bx), \quad (x,y)\in \mathbb R^2, \tag{1} \] where \(p(x)\) is a polynomial and \(b\neq 0\) is a constant. The case \(p(x)=x^2+a\) corresponds to the well-known and much studied quadratic Hénon map. The study of the dynamics of maps (1) is of prime interest since by the result of Friedland and Milnor any Cremona map (constant Jacobian polynomial map with polynomial inverse) is conjugate to a composition of generalized Hénon maps, or else has trivial dynamics. Much attention is given to the study of the dynamics of the maps (1), where \(p(x)=\pm x^d+ O(x^{d-2})\). General properties (symmetries, low period orbits, Birkhoff normal form, bounded orbits, and anti-integrable limit) are discussed. Area-preserving cubic maps of the form (1) (\(p(x)=\pm x^3+lx+k, b=1\)) are subjected to an extended discussion. The study of these maps includes fixed points and low period orbits, Birkhoff normal form, and anti-integrable limit. When the critical points of \(p(x)\) are nondegenerate, \(h\) is shown to have an anti-integrable limit and a domain in the parameter \((l,k)\)-space for which there is a hyperbolic horseshoe. The bifurcations of fixed points and low (two and three) period orbits are discussed in detail. In particular, curves in the parameter plane for codimension-one bifurcations are obtained. A number of codimension-two bifurcation is found and classified. The paper also includes a short, self-contained review of results on invertible polynomial maps of the plane.
scientific article; zbMATH DE number 1538314
scientific article