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Green currents for quasi-algebraically stable meromorphic self-maps of \(\mathbb{P}^k\). (English) Zbl 1297.37023

Let \(f: \mathbb P^k \to \mathbb P^k\) be a dominant meromorphic map. The map \(f\) is algebraically stable if the algebraic degree of \(f^{\circ n}\) equals the algebraic degree of \(f\), \(d(f)\), to the power \(n\). For algebraically stable maps many things are known. For instance Sibony proved that if \(f\) is algebraically stable and \(d(f)>1\), then there exists a unique current \(T\), called the Green current of \(f\), which satisfies \(f^\ast(T)=d(f)T\) and essentially describes the dynamics of \(f\). For non-algebraically stable mappings, not much is known. Some classes of non algebraically stable maps have been studied by Bonifant and Fornæss, while Diller and Favre, and Favre, Jonsson and later together with Boucksom, have studied non algebraically stable maps in dimension two (references are in the bibliography). Other particular cases are studied by Hasselblatt and Propp and by Bedford and Kim.
In the paper under review the author studies a new class of non algebraically stable maps: the class of “quasi-algebraically stable maps”. A map is quasi-algebraically stable provided there exist an integer \(n_0\geq 1\), a homogeneous polynomial \(H\) and a sequence \(\{F_n\}\) of liftings to \(\mathbb C^{k+1}\) of the sequence of iterates \(\{f^{\circ n}\}\) of \(f\) such that \(F_n=F_1\circ F_{n-1}\) for \(n\leq n_0\) and \(F_n=F_1\circ F_{n-1}/H\circ F_{n-n_0-1}\) for \(n>n_0\).
Let \(f\) be quasi-algebraically stable. Let \(\lambda\) denote the first dynamical degree of \(f\) and let \(h\) be the degree of \(H\). It is known that \(\lambda\) is a root of the polynomial \(P(t)= t^{n_0+1}-dt^{n_0}+h\). Assume that \(\lambda\) is a simple root of \(P(t)\) and \(\lambda>1\). Then the main result of the paper is the following: the upper semicontinuous regularization of \(\limsup_{n\to\infty}\log\|F_n\|/d(f^n)\) exists and defines a plurisubharmonic function \(u\) on \(\mathbb C^{k+1}\). The positive closed \((1,1)\)-current \(T\) on \(\mathbb P^k\) defined by \(u\) satisfies the functional equation \(f^\ast(T)=\lambda T+(d(f)-\lambda)/h\cdot [H=0]\). Moreover, the support of \(T\) is contained in the Julia set of \(f\), which is thus not empty.
The paper ends with a sufficient criterion for a map to be quasi-algebraically stable in terms of degree lowering hypersurfaces, and with the construction of a new family of such maps.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32U40 Currents
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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