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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 \(\mu\).

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
54H20 Topological dynamics (MSC2010)
Full Text: DOI Numdam EuDML
[1] Bedford, E.; Smillie, J., Polynomial diffeomorphisms of \(\mathbb{C}^2\). III. ergodicity, exponents and entropy of the equilibrium measure, Math. Ann., 294, 395-420, (1992) · Zbl 0765.58013
[2] Berteloot, F.; Dupont, C., Une caractérisation des endomorphismes de Lattès par leur mesure de Green, Comment. Math. Helv., 80, 433-454, (2005) · Zbl 1079.37039
[3] Berteloot, F.; Mayer, V., Rudiments de dynamique holomorphe, 7, (2001), Société Mathématique de France, Paris · Zbl 1051.37019
[4] Briend, J.-Y., Exposants de Liapounoff et points périodiques d’endomorphismes holomorphes de \(\mathbb{C} \mathbb{P}^k, (1997)\)
[5] Briend, J.-Y.; Duval, J., Exposants de liapounoff et distribution des points périodiques d’un endomorphisme de \(\mathbb{C} \mathbb{P}^k,\) Acta Math., 182, 143-157, (1999) · Zbl 1144.37436
[6] Briend, J.-Y.; Duval, J., Deux caractérisations de la mesure d’équilibre d’un endomorphisme de \(\mathbb{P}^k(\mathbb{C}),\) IHES, Publ. Math., 93, 145-159, (2001) · Zbl 1010.37004
[7] Brin, M.; Katok, A., On local entropy, 1007, (1983), Lect. Notes in Math., Springer Verlag · Zbl 0533.58020
[8] Cantat, S., Dynamique des automorphismes des surfaces K3, Acta Math., 187, 1-57, (2001) · Zbl 1045.37007
[9] Carleson, L.; Gamelin, T. W., Complex dynamics, (1993), Springer-Verlag · Zbl 0782.30022
[10] Dinh, T.-C., Suites d’applications méromorphes multivaluées et courants laminaires, J. Geom. Anal., 15, 207-227, (2005) · Zbl 1085.37039
[11] Dujardin, R., Laminar currents and birational dynamics · Zbl 1099.37037
[12] Dujardin, R., Laminar currents in \(\mathbb{P}^2,\) Math. Ann., 325, 745-765, (2003) · Zbl 1021.37018
[13] Dujardin, R., Sur l’intersection des courants laminaire, Publ. Mat., 48, 107-125, (2004) · Zbl 1048.32021
[14] Fornæss, J. E.; Sibony, N., Complex dynamics in higher dimension I, Astérisque, 222, 201-231, (1994) · Zbl 0813.58030
[15] Fornæss, J. E.; Sibony, N., Complex dynamics in higher dimensions, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, 131-186, (1994) · Zbl 0811.32019
[16] Fornæss, J. E.; Sibony, N., Complex dynamics in higher dimension II, Ann. Math. Studies, 137, 135-182, (1995) · Zbl 0847.58059
[17] Fornæss, J. E.; Sibony, N., Hyperbolic maps on \(\mathbb{P}^2,\) Math. Ann., 311, 305-333, (1998) · Zbl 0928.37006
[18] Katok, A.; Hasselblatt, B., Introduction to the modern theory of dynamical systems, 54, (1995), Cambridge University Press · Zbl 0878.58020
[19] Kozlovski, O. S., An integral formula for topological entropy of \(C^{∞ }\) maps, Ergodic Theory Dynam. Systems, 18, 405-424, (1998) · Zbl 0915.58058
[20] Pugh, C.; Shub, M., Ergodic attractors, Trans. Amer. Math. Soc., 312, 1-54, (1989) · Zbl 0684.58008
[21] Ruelle, D., An inequality for the entropy of differentiable maps, Bol. Soc. Brasil Mat., 9, 83-87, (1978) · Zbl 0432.58013
[22] de Thélin, H., Sur la laminarité de certains courants, Ann. Sci. Ecole Norm. Sup., 37, 304-311, (2004) · Zbl 1061.32005
[23] de Thélin, H., Un phénomène de concentration de genre, Math. Ann., 332, 483-498, (2005) · Zbl 1076.37037
[24] Ueda, T., Critical orbits of holomorphic maps on projective spaces, J. Geom. Anal., 8, 319-334, (1998) · Zbl 0957.32009
[25] Walters, P., An introduction to ergodic theory, (1982), Springer, Berlin Heidelberg New York · Zbl 0475.28009
[26] Yomdin, Y., Volume growth and entropy, Israel J. Math., 57, 285-300, (1987) · Zbl 0641.54036
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