zbMATH — the first resource for mathematics

Polynomial diffeomorphisms of \(\mathbb{C}^ 2\). IV: The measure of maximal entropy and laminar currents. (English) Zbl 0792.58034
For the study of invertible, holomorphic dynamical systems, the simplest examples with interesting behavior are the polynomial diffeomorphisms of \(\mathbb{C}^ 2\). The paper under review is the fourth in a series initiated by two of the present authors [see E. Bedford and J. Smillie, ibid. 103, No. 1, 69-99 (1991; Zbl 0721.58037), J. Am. Math. Soc. 4, No. 4, 657-679 (1991; Zbl 0744.58068), Math. Ann. 294, No. 3, 395-420 (1992; Zbl 0765.58013)].
Corresponding to a Julia set in one dimension, for diffeomorphisms of \(\mathbb{C}^ 2\) three analogs are defined. The techniques of Fatou and Julia used in one dimension are based on Montel’s theory of normal families and do not readily generalize to higher dimensions. Later, it was seen that potential theory is an alternative tool to prove basic facts of Fatou- Julia theory. Potential theory has a natural extension to several complex variables.
The authors combine the methods of potential theory with tools from ergodic theory, especially Pesin’s theory of non-uniform hyperbolicity. The first crucial result is the following: The harmonic measure \(\mu\) is the unique measure of maximal entropy for a polynomial diffeomorphism of \(\mathbb{C}^ 2\).
It is shown that are sets, called Pesin boxes, on which \(\mu\) has a local product structure and their union has full \(\mu\) measure. For doing this, the holonomy map along the stable/unstable manifolds is studied; the conditional measures of \(\mu\) are preserved by the holonomy map.
Next, the primary tools are the theory of currents and the Ahlfors covering theorem. It is proved that \(\mu^ \pm\) are laminar currents, where the harmonic measure \(\mu\) and the currents \(\mu^ \pm\) are related by \(\mu = \mu^ + \wedge \mu^ -\). The holonomy map preserves the slices of \(\mu^ +\). Applications to the study of saddle points and to real Hénon mappings are given.
An appendix outlines an alternative sequence in which the results of this paper can be obtained. This alternate approach starts with Pesin theory and then proceeds to the theory of currents. The main difference is that the use of the methods of entropy theory is delayed until the end.

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
58C35 Integration on manifolds; measures on manifolds
58A25 Currents in global analysis
Full Text: DOI EuDML arXiv
[1] [Be] Bedford, E.: Iteration of polynomial automorphisms ofC 2. In: Satake, I. (ed.) Proceedings of the I.C.M. 1990. Kyoto, Japan, pp. 847-858. Berlin Heidelberg New York: Springer 1991 · Zbl 0746.58040
[2] [BS1] Bedford, E., Smillie, J.: Polynomial diffeomorphisms ofC 2: Currents, equilibrium measure and hyperbolicity. Invent. Math.87, 69-99 (1990) · Zbl 0721.58037
[3] [BS2] Bedford, E., Smillie, J.: Fatou-Bieberbach domains arising from polynomial automorphisms. Indiana Univ. Math. J.40, 789-792 (1991) · Zbl 0739.32027 · doi:10.1512/iumj.1991.40.40035
[4] [BS3] Bedford, E., Smillie, J.: Polynomial diffeomorphisms ofC 2 II: Stable manifolds and recurrence. J. Am. Math. Soc.4 (No. 4), 657-679 (1991) · Zbl 0744.58068
[5] [BS4] Bedford, E., Smillie, J.: Polynomial diffeomorphisms ofC 2 III: Ergodicity, exponents and entropy of the equilibrium measure. Math. Ann.294, 395-420 (1992) · Zbl 0765.58013 · doi:10.1007/BF01934331
[6] [BT1] Bedford, E., Taylor, B.A.: Fine topology, ?hilov boundary and (dd c ) n J. Funct. Anal.72, 225-251 (1987) · Zbl 0677.31005 · doi:10.1016/0022-1236(87)90087-5
[7] [BT2] Bedford, E., Taylor, B.A.: Positive, closed currents and complex cycles. Unpublished manuscript (1988)
[8] [Bo] Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms, (Lect. Notes, Math. vol. 470) Berlin Heidelberg New York: Springer 1975 · Zbl 0308.28010
[9] [Br] Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat.,6, 103-144 (1965) · Zbl 0127.03401 · doi:10.1007/BF02591353
[10] [C] Carleson, L.: Complex dynamics. UCLA course notes (1990)
[11] [CFS] Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic Theory. Berlin Heidelberg New York: Springer 1982 · Zbl 0493.28007
[12] [D] Demailly, J.-P.: Courants positifs extr?maux et conjecture de Hodge. Invent. Math.69, 347-374 (1982) · Zbl 0488.58001 · doi:10.1007/BF01389359
[13] [EL] Eremenko, A., Lyubich, M.: Dynamics of analytic transformations. Leningr. J. Math.1, 1-70 (1989) · Zbl 0712.58036
[14] [FHY] Fathi, A., Herman, M., Yoccoz, J.-C.: A proof of Pesin’s stable manifold theorem. In: Palis, J. (ed.) Geometric Dynamics. Berlin Heidelberg New York Springer: (Lect. Notes Math., vol. 1007 pp. 177-215)
[15] [FS] Forn?ss, J.-E. Sibony, N.: Complex H?non mappings inC 2 and Fatou Bieberbach domains. Duke Math. J.65, 345-380 (1992) · Zbl 0761.32015 · doi:10.1215/S0012-7094-92-06515-X
[16] [FM] Friedland, S., Milnor, J.: Dynamical properties of plane polynomial automorphisms. Ergodic Theory Dyn. Syst.9, 67-99 (1989) · Zbl 0651.58027 · doi:10.1017/S014338570000482X
[17] [G] Gromov, M.: On the entropy of holomorphic maps. Unpublished manuscript (1980)
[18] [Ha] Hayman, W.: Meromorphic Functions. Oxford: Oxford University Press, 1964 · Zbl 0115.06203
[19] [H] Hubbard, J.H.: The H?non mapping in the complex domain, In: Barnsley, M., Demko, S. (eds.) Chaotic Dynamics and Fractals, pp. 101-111. New York: Academic Press 1986
[20] [HO] Hubbard, J.H., Oberste-Vorth, R.: H?non mappings in the complex domain (in preparation) · Zbl 0839.54029
[21] [HP] Hubbard, J.H., Papadopol, P.: Superattractive fixed points inC n . (Preprint) · Zbl 0858.32023
[22] [K] Katok, A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math., Inst. Hautes ?tud. Sci.51, 137-174 (1980) · Zbl 0445.58015 · doi:10.1007/BF02684777
[23] [Kl] Klimek, M.: Pluripotential theory. Oxford: Oxford University Press 1991 · Zbl 0742.31001
[24] [Kr] Krieger, W.: On entropy and generators of measure-preserving transformations. Trans. Am. Math. Soc.119, 453-464 (1970) · Zbl 0204.07904 · doi:10.1090/S0002-9947-1970-0259068-3
[25] [Le] Ledrappier, F.: Some properties of absolutely continuous invariant measures on an interval. Ergodic Theory Dyn. Syst.1, 77-93 (1981) · Zbl 0487.28015 · doi:10.1017/S0143385700001176
[26] [LS] Ledrappier, F., Strelcyn, J.-M.: A proof of the estimation from below in Pesin’s entropy formula. Ergodic Theory Dyn. Syst.2, 203-219 (1982) · Zbl 0533.58022 · doi:10.1017/S0143385700001528
[27] [LY] Ledrappier, F., Young, L.-S.: The metric entropy of diffeomorphisms. I. Ann. Math.122, 509-539 (1985); II. Ann. Math.122, 540-574 (1985) · Zbl 0605.58028 · doi:10.2307/1971328
[28] [L] Lelong, P.: El?ments extr?maux sur le c?ne des courants positifs ferm?s. In: S?minaire P. Lelong (Analyse). Ann?e, 1971-1972. (Lect. Notes Math., vol. 332) Berlin Heidelberg New York: Springer 1972
[29] [Lyu1] Lyubich, M.: Entropy of analytic endomorphisms of the Riemannian sphere. Funct. Anal. Appl.15, 300-302 (1981) · Zbl 0483.30032 · doi:10.1007/BF01106165
[30] [Lyu2] Lyubich, M.: Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dyn. Syst.3, 351-385 (1983) · Zbl 0537.58035
[31] [Ma] Ma??, R.: On the uniqueness of the maximizing measure for rational maps. Bol. Soc. Bras. Math.14, 27-43 (1983) · Zbl 0568.58028 · doi:10.1007/BF02584743
[32] [Mak] Makarov, N.G.: On the distortion of boundary sets under conformal mappings. Proc. Lond. Math. Soc.51, 369-384 (1985) · Zbl 0573.30029 · doi:10.1112/plms/s3-51.2.369
[33] [Man] Manning, A.: The dimension of the maximal measure for a polynomial map. Ann. Math.119, 425-430 (1984) · Zbl 0551.30021 · doi:10.2307/2007044
[34] [Mi] Milnor, J.: Non-expansive H?non maps. Adv. Math.69, 109-114 (1988) · Zbl 0647.58043 · doi:10.1016/0001-8708(88)90062-X
[35] [Mo] Morse, A.P.: Perfect blankets. Trans. Am. Math. Soc.61, 418-442 (1947) · Zbl 0031.38702 · doi:10.1090/S0002-9947-1947-0020618-0
[36] [N] Nevanlinna, R.: Analytic Functions. Berlin Heidelberg New York: Springer 1970 · Zbl 0199.12501
[37] [Ne] Newhouse, S.: Continuity properties of entropy. Ann. Math.129, 215-235 (1989) · Zbl 0676.58039 · doi:10.2307/1971492
[38] [OW] Ornstein, D., Weiss, B.: Statistical properties of chaotic systems. Bull. Am. Math. Soc.24, 11-115 (1991) · Zbl 0718.58038 · doi:10.1090/S0273-0979-1991-15953-7
[39] [PdM] Palis, J., de Malo, W.: Geometric Theory of Dynamical Systems. Berlin Heidelberg New York: Springer 1982
[40] [Pe] Pedersen, P.: Personal communication
[41] [P1] Pesin, Y.: Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv.32, 55-114 (1977) · Zbl 0383.58011 · doi:10.1070/RM1977v032n04ABEH001639
[42] [P2] Pesin, Y.: Description of the ?-partition of a diffeomorphism with invariant smooth measure. Mat. Zametki21, 29-44 (1977)
[43] [Pr] Przytycki, F.: Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map. Invent. Math.80, 161-179 (1985) · Zbl 0569.58024 · doi:10.1007/BF01388554
[44] [PS] Pugh, C., Shub, M.: Ergodic attractors. Trans. Am. Math. Soc.312, 1-54 (1989) · Zbl 0684.58008 · doi:10.1090/S0002-9947-1989-0983869-1
[45] [Ro1] Rokhlin, V.A.: On the fundamental ideas of measure theory, Mat. Sb.25 (67), 107-150 (1949) AMS (1952)
[46] [Ro2] Rohlin, V.A.: Lectures on entropy theory of measure-preserving transformation. Russ. Math. Surv.22 (1967)
[47] [R1] Ruelle, D.: An inequality for the entropy of differential maps. Bol. Soc. Bras. Mat.9, 83-87 (1978) · Zbl 0432.58013 · doi:10.1007/BF02584795
[48] [R2] Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. Math., Inst. Hautes ?tud. Sci.50, 275-306 (1979)
[49] [RS] Ruelle, D., Sullivan, D.: Currents, flows, and diffeomorphisms. Topology14, 319-327 (1975) · Zbl 0321.58019 · doi:10.1016/0040-9383(75)90016-6
[50] [Si] Sibony, N.: Course at UCLA. Unpublished manuscript
[51] [S] Smillie, J.: The entropy of polynomial diffeomorphisms ofC 2. Ergodic Theory Dyn. Syst.10, 823-827 (1990) · Zbl 0695.58023 · doi:10.1017/S0143385700005927
[52] [Su] Sullivan, D.: Dynamical study of foliated and complex manifolds. Invent. Math.36, 225-255 (1976) · Zbl 0335.57015 · doi:10.1007/BF01390011
[53] [T] Tortrat, P.: Aspects potentialistes de l’it?ration des polyn?mes. In: S?minaire de Th?orie du Potentiel Paris, No. 8. (Lect. Notes Math., vol. 1235) Berlin Heidelberg New York: Springer 1987
[54] [W] Wu, H.: Complex stable manifolds of holomorphic diffeomorphisms. Indiana Univ. Math. J. (to appear) · Zbl 0811.58009
[55] [Yg] Young, L.-S.: Dimension, entropy and Lyapunov exponents. Ergodic Theory Dyn. Syst.2, 109-124 (1982) · Zbl 0523.58024 · doi:10.1017/S0143385700009615
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.