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On saddle measures. (Sur la construction de mesures selles.) (French) Zbl 1100.37029
Let \(f : \mathbb{P}^2 \to \mathbb{P}^2\) be a holomorphic endomorphism of the complex projective plane. We note \(T = \lim_{n \to +\infty} {1 \over d^n} {f^n}^*\omega\) the Green current and \(\mu = T \wedge T\) the Green measure of \(f\), where \(\omega\) is the Fubini-Study \((1,1)\) form and \(d \geq 2\) is the algebraic degree of \(f\). Let \(L\) be a line in \(\mathbb{P}^2\) and \(S\) be a cluster value of the sequence of positive \((1,1)\) currents \(S_m = {1 \over m} \sum_{i=0}^{m-1} [f^i(L)]/d^i\). The measure \(\nu = T \wedge S\) is invariant by \(f\). If \(f\) is strongly hyperbolic, J. E. Fornaess and N. Sibony [Math. Ann. 311, 305–333 (1998; Zbl 0928.37006)] proved in particular that the Lyapounov exponents at every point \(x\) of the support of \(\nu\) satisfy \(\lambda_1(x) < 0 < \lambda_2(x)\).
The author generalizes this result to every endomorphism \(f\) of \(\mathbb{P}^2\) in a nonuniform setting. He proves that if \(\nu\) does not charge any algebraic curve, then \(\lambda_1(x) \leq 0\) for \(\nu\)-generic points outside the support of \(\mu\). He proves also (without any condition on \(\nu\)) that the metric entropy \(h(\nu)\) is bounded below by \(\log d\) and that \(\log \sqrt d \leq \lambda_2(x)\) for \(\nu\)-generic points. These results give a nice understanding of the dynamics of \(f\) outside the support of \(\mu\).

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
54H20 Topological dynamics (MSC2010)
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