Real polynomial diffeomorphisms with maximal entropy: tangencies. (English) Zbl 1084.37034

Bedford, Eric; Smillie, John

Ann. Math. (2) 160, No. 1, 1-26 (2004).

The authors discuss several questions concerning diffeomorphisms on \(\mathbb{R}^2\) pointing out that a “model family” is that of Hénon \(\{f_{a,b}\}:f_{a,b} (x,y)=(a-by-x^2,x)\) with \(b\neq 0\). If \(b\neq 0\), then dynamics of \(f_{a,0}\) reduces to that of quadratic maps \(f_a(x)=a-x^2\) playing a central role in the theory of one-dimensional dynamical systems. The paper consists of the introduction (presenting an instructive outline of historical roots of investigated problems and information on the contents of the paper) and five sections having titles well suggesting the subjects of them: Background, The maximal entropy condition and its consequences, Finiteness of singular points, Hyperbolicity and tangencies, One-sided points.
Let us mention at least some examples of interesting results: Suppose that a diffeomorphism has a maximal entropy. Then the set of points with bounded orbits is a Cantor set (Theorem 5.2). For a real, polynomial mapping \(f\) of maximal entropy: \(f\) is not hyperbolic if and only if there are saddle points \(p\) and \(q\) such that the stable manifold \(W^s(p)\) intersects the unstable manifold \(W^u(q)\) tangentially (a part of Theorem 4.4). Existence of stable and unstable one-sided points for hyperbolic mappings is proved (Theorem 5.1).
An important authors’ remark is expressed by the two first sentences of Section 1 (Background): Despite the fact that we study real polynomial diffeomorphisms, the proofs of the results of this paper depend on the theory of complex polynomial diffeomorphisms. In particular, the theory of quasi-hyperbolicity which lies at the heart of much of what we do is a theory of complex polynomial diffeomorphisms.

Reviewer: Andrzej Pelczar (Kraków)

MSC:

37E30	Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37B40	Topological entropy
37C05	Dynamical systems involving smooth mappings and diffeomorphisms
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37C75	Stability theory for smooth dynamical systems
37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets

Keywords:

topological conjugacy; dynamics of diffeomorphisms; hyperbolicity; tangencies; one-sided stably and unstably points; preserving and reserving orientation mappings