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Hénon-like mappings in \(\mathbb C^2\). (English) Zbl 1064.37035

A Hénon-like map is an injective holomorphic (possibly non algebraic) map defined on a neighborhood of the unit bidisc \(\mathbb{B}\) which “looks like” a Hénon map. More precisely, one require that \(f( \{|{z}|=1, |{w}| < 1 \} ) \cap \overline {\mathbb{B}} = \emptyset\) and \(f( \overline {\mathbb{B}}) \cap \partial \mathbb{B} \subset \{|{z}|=1, |{w}| < 1 \}\). The dynamics of Hénon maps have been deeply studied in the 90’s by J. E. Fornaess and N. Sibony [Duke Math. J. 65, 345–380 (1992; Zbl 0761.32015)] and E. Bedford, M. Lyubich and J. Smillie [Invent. Math. 112, 77–125 (1993; Zbl 0792.58034)]. Such maps admit a unique measure \(\mu\) of maximal entropy, which is mixing, weakly hyperbolic and describes the asymptotic distribution of periodic saddle points.
In the article in review, the author shows that the Hénon-like maps share the same properties. The construction of \(\mu\) follows the Hénon situation: \(\mu\) appears as the product \(T^+ \wedge T^-\) of unstable and stable \((1,1)\) positive closed currents. These ones are not related with Green functions but are built by means of a graph transform method. As in the Hénon case, \(T^+\) and \(T^-\) are laminated by the unstable and stable foliations of the dynamical system. Finally, the author gives some transcendantal examples of Hénon like maps (e.g., \((z,w) \mapsto (aw+\lambda e^z,az)\) for \(a\) small enough).

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
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