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The bifurcation measure has maximal entropy. (English) Zbl 1437.37061
By the analogy between the dynamics of an endomorphism of \(\mathbb{P}^k\) and bifurcation in a holomorphic family of rational maps, the main goal of the paper is to show that the bifurcation measure has maximal entropy (Theorem B).
For this purpose, the authors first introduce the notions of bifurcation entropy and metric bifurcation entropy, based on the concept of a \((d_n,\epsilon)\)-separated set, where \(d_n\) is the so-called \(n\)-bifurcation distance on the parameter space. One of the key ingredients is a generalization of Yomdin’s bound of the volume of the image of a dynamical ball.
The authors give an alternative proof of the computation of the entropy of the Green measure of Hénon maps, and define and compute the entropy of the trace measure of the Green currents of a holomorphic endomorphism of \(\mathbb{P}^k\).
MSC:
37F80 Higher-dimensional holomorphic and meromorphic dynamics
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F46 Bifurcations; parameter spaces in holomorphic dynamics; the Mandelbrot and Multibrot sets
32U40 Currents
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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[1] Astorg, M.; Gauthier, T.; Mihalache, N.; Vigny, G., Collet, Eckmann and the bifurcation measure, Inventiones Mathematicae, 217, 749-797 (2019) · Zbl 07089444
[2] Bassanelli, G.; Berteloot, F., Bifurcation currents in holomorphic dynamics on ℙ^k, Journal für die Reine und Angewandte Mathematik, 608, 201-235 (2007) · Zbl 1136.37025
[3] Bassanelli, G.; Berteloot, F., Distribution of polynomials with cycles of a given multiplier, Nagoya Mathematical Journal, 201, 23-43 (2011) · Zbl 1267.37049
[4] Bedford, E.; Lyubich, M.; Smillie, J., Distribution of periodic points of polynomial diffeomorphisms ofC^2, Inventiones Mathematicae, 114, 277-288 (1993) · Zbl 0799.58039
[5] Bedford, E.; Smillie, J., Polynomial diffeomorphisms ofC^2: currents, equilibrium measure and hyperbolicity, Inventiones Mathematicae, 103, 69-99 (1991) · Zbl 0721.58037
[6] Bedford, E.; Smillie, J., Polynomial diffeomorphisms ofC^2. III. Ergodicity, exponents and entropy of the equilibrium measure, Mathematische Annalen, 294, 395-420 (1992) · Zbl 0765.58013
[7] Berteloot, F.; Dupont, C., Une caractérisation des endomorphismes de Lattès par leur mesure de Green, Commentarii Mathematici Helvetici, 80, 433-454 (2005) · Zbl 1079.37039
[8] Briend, J-Y; Duval, J., Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme deCP^k, Acta Mathematica, 182, 143-157 (1999) · Zbl 1144.37436
[9] Briend, J-Y; Duval, J., Deux caractérisations de la mesure d’équilibre d’un endomorphisme de P^k(C), Publications Mathématiques. Institut des Hautes Études Scientifiques, 93, 145-159 (2001) · Zbl 1010.37004
[10] Brin, M.; Katok, A., On local entropy, in Geometric Dynamics (Rio de Janeiro, 1981), 30-38 (1983), Berlin: Springer, Berlin · Zbl 0533.58020
[11] Buff, X.; Epstein, A., Bifurcation measure and postcritically finite rational maps, Complex Dynamics: Families and Friends, 491-512 (2009), Wellesley, MA: A K Peters, Wellesley, MA · Zbl 1180.37056
[12] Burguet, D., A proof of Yomdin-Gromov’s algebraic lemma, Israel Journal of Mathematics, 168, 291-316 (2008) · Zbl 1169.14038
[13] Demarco, L., Dynamics of rational maps: a current on the bifurcation locus, Mathematical Research Letters, 8, 57-66 (2001) · Zbl 0991.37030
[14] De Thelin, H., Un phénomène de concentration de genre, Mathematische Annalen, 332, 483-498 (2005) · Zbl 1076.37037
[15] De Thélin, H., Sur la construction de mesures selles, Université de Grenoble. Annales de l’Institut Fourier, 56, 337-372 (2006) · Zbl 1100.37029
[16] H. De Thélin and G. Vigny, Entropy of meromorphic maps and dynamics of birational maps, Memoires de la Société Mathématique de France 122 (2010). · Zbl 1214.37004
[17] Dinh, T-C, Decay of correlations for Hénon maps, Acta Mathematica, 195, 253-264 (2005) · Zbl 1370.37087
[18] Dinh, T-C, Attracting current and equilibrium measure for attractors on ℙ^k, Journal of Geometric Analysis, 17, 227-244 (2007) · Zbl 1139.37032
[19] Dinh, T-C; Sibony, N., Dynamics of regular birational maps in ℙ^k, Journal of Functional Analysis, 222, 202-216 (2005) · Zbl 1067.37055
[20] Dujardin, R., Fatou directions along the Julia set for endomorphisms of ℂℙ^k, Journal de Mathematiques Pures et Appliquees, 98, 591-615 (2012) · Zbl 1333.37024
[21] Dujardin, R., The supports of higher bifurcation currents, Annales de la Faculté des Sciences de Toulouse. Mathématiques, 22, 445-464 (2013) · Zbl 1314.37032
[22] Dujardin, R., Bifurcation currents and equidistribution in parameter space, in Frontiers in Complex Dynamics, 515-566 (2014), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 1405.37003
[23] Dujardin, R.; Favre, C., Distribution of rational maps with a preperiodic critical point, American Journal of Mathematics, 130, 979-1032 (2008) · Zbl 1246.37071
[24] Favre, C.; Gauthier, T., Distribution of postcritically unite polynomials, lsrael, Journal of Mathematics, 209, 235-292 (2015) · Zbl 1352.37202
[25] Favre, C.; Rivera-Letelier, J., Equidistribution quantitative des points de petite hauteur sur la droite projective, Mathematische Annalen, 335, 311-361 (2006) · Zbl 1175.11029
[26] Fornœss, J. E.; Sibony, N., Complex dynamics in higher dimension, Modern Methods in Complex Analysis (Princeton, NJ, 1992), 135-182 (1995), Princeton, NJ: Princeton University Press, Princeton, NJ
[27] Gauthier, T., Strong bifurcation loci of full Hausdorff dimension, Annales Scientifiques de l’École Normale Supérieure, 45, 947-984 (2012) · Zbl 1326.37036
[28] Gauthier, T.; Okuyama, Y.; Vigny, G., Hyperbolic components of rational maps: Quantitative equidistribution and counting, Commentarii Mathematici Helvetici, 94, 347-398 (2019) · Zbl 1431.37041
[29] Gauthier, T.; Vigny, G., Distribution of postcritically unite polynomials II: Speed of convergence, Journal of Modern Dynamics, 11, 57-98 (2017) · Zbl 1419.37040
[30] Gauthier, T.; Vigny, G., Distribution of postcritically unite polynomials III: Combinatorial continuity, Fundamenta Mathematicae, 244, 17-48 (2019) · Zbl 1432.37074
[31] Graczyk, J.; Światek, G., Lyapunov exponent and harmonic measure on the boundary of the connectedness locus, International Mathematics Research Notices, 16, 7357-7364 (2015) · Zbl 1347.37088
[32] Gromov, M., Entropy, homology and semialgebraic geometry, Astérisque, 145-146, 225-240 (1987) · Zbl 0611.58041
[33] Gromov, M., On the entropy of holomorphic maps, L’Enseignement Mathématique, 49, 217-235 (2003) · Zbl 1080.37051
[34] Inou, H.; Mukherjee, S., Non-landing parameter rays of the multicorns, Inventiones Mathematicae, 204, 869-893 (2016) · Zbl 1407.37074
[35] Katok, A., Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Institut des Hautes Études Scientifiques. Publications Mathématiques, 51, 137-173 (1980) · Zbl 0445.58015
[36] Levin, G., On the theory of iterations of polynomial families in the complex plane, Journal of Soviet Mathematics, 52, 3512-3522 (1990) · Zbl 0716.30017
[37] Lyubich, M., Investigation of the stability of the dynamics of rational functions, Teoriya Funktsiĭ, Funktsional’nyĭ Analiz i ikh Prilozheniya, 42, 72-91 (1984) · Zbl 0572.30023
[38] Mañé, R.; Sad, P.; Sullivan, D., On the dynamics of rational maps, Annales Scientifiques de l’École Normale Supérieure, 16, 193-217 (1983) · Zbl 0524.58025
[39] Yomdin, Y., Volume growth and entropy, Israel Journal of Mathematics, 57, 285-300 (1987) · Zbl 0641.54036
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