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

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.

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 |