# zbMATH — the first resource for mathematics

Polynomial diffeomorphisms of $${\mathbb{C}}^2$$. V: Critical points and Lyapunov exponents. (English) Zbl 1004.37027
Over the last 10 years, E. Bedford and J. Smillie (in part with M. Lyubich) have published a steady stream of works on Polynomial Diffeomorphisms of $$\mathbb{C}^2$$: For Parts I–IV, see E. Bedford and J. Smillie, Currents, equilibrium measure and hyperbolicity. Invent. Math. 103, No. 1, 69-99 (1991; Zbl 0721.58037); Stable manifolds and recurrence. J. Am. Math. Soc. 4, No. 4, 657-679 (1991; Zbl 0744.58068); Ergodicity, exponents and entropy of the equilibrium measure. Math. Ann. 294, No. 3, 395-420 (1992; Zbl 0765.58013); The Measure of maximal entropy and laminar currents. Invent. Math. 112, No. 1, 77-125 (1993; Zbl 0792.58034).
In this paper, Part V, the authors give an integral formula for the Lyapunov exponents of a dynamically non-trivial polynomial automorphism, associated with the unique invariant measure of maximal entropy.
For Parts VI–VIII, see the following review (Zbl 1004.37034).
Main Theorem: For any $$t>0$$, the Lyapunov exponent of a polynomial automorphism $$f$$ with degree $$d$$, is estimated as follows: $\Lambda_\mu(f)= \log d+\int_{t\leq G^+<td} G^+d\mu_c^-,$ where $$\mu_c^-$$ is a critical measure $$G^+$$, Green function associated to $$f$$.
This is analogous to the Brolin-Manning formula for a polynomial map $$p$$ of $$\mathbb{C}$$ with degree $$d$$: Manning-Przytycki Result: The Lyapunov exponent is as follows: $\lambda_\nu (p)=\log d+\sum_iG(c_i),$ where $$\nu$$ is Brolin measure, $$G$$ Green function of the Julia set of $$p$$, and $$\{c_i\}$$ the set of critical points.
Further information on this theory will be available on the World Wide Web at:
http://www.math.sunysb.edu/dynamics/preprints/preprint.html.

##### MSC:
 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 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
Full Text:
##### References:
 [1] Bedford, E. and Smillie, J. Polynomial diffeomorphisms ofC 2: Currents, equilibrium measure and hyperbolicity,Invent. Math.,87, 69–99, (1990). · Zbl 0721.58037 [2] Bedford, E. and Smillie, J. Polynomial diffeomorphims ofC 2, III: Ergodicity, exponents and entropy of the equilibrium measure,Math. Ann.,294, 395–420, (1992). · Zbl 0765.58013 · doi:10.1007/BF01934331 [3] Bedford, E. and Smillie, J. Polynomial diffeomorphisms ofC 2, VI:Connectivity of J. Annals of Math.,148, 695–735, (1998). · Zbl 0916.58022 [4] Bedford, E. and Smillie, J. Polynomial diffeomorphisms ofC 2, VII: Hyperbolicity and external rays.Annales Scientifiques de L’Ecole Normale Supérieure, to appear. · Zbl 0952.37008 [5] Bedford, E., Lyubich, M., and Smillie, J. Polynomial diffeomorphims ofC 2, IV: The measure of maximal entropy and laminar currents,Invent. Math.,112, 77–125, (1993). · Zbl 0792.58034 · doi:10.1007/BF01232426 [6] Bedford, E., Lyubich, M., and Smillie, J. Distribution of periodic points of polynomial diffeomorphisms ofC 2,Invent. Math.,114, 277–288, (1993). · Zbl 0799.58039 · doi:10.1007/BF01232671 [7] Benedicks, M. and Carleson, L. The dynamics of the Hénon map,Ann. Math.,133, 73–169, (1991). · Zbl 0724.58042 · doi:10.2307/2944326 [8] Benedicks, M. and Young, L-S. Sinai-Bowen-Ruelle measures for certain Hénon maps,Invent. Math.,112, 541–576, (1993). · Zbl 0796.58025 · doi:10.1007/BF01232446 [9] Brolin, H. Invariant sets under iteration of rational functions,Ark. Mat.,6, 103–144, (1965). · Zbl 0127.03401 · doi:10.1007/BF02591353 [10] Chirka, E.Complex Analytic Sets, Kluwer, (1985). · Zbl 0781.32011 [11] Douady, A. and Hubbard, J.H. Itération des polynômes quadratiques complexes,C.R. Acad. Sci. Paris Série I,294, 123–126, (1982). · Zbl 0483.30014 [12] Friedland, S. and Milnor, J. Dynamical properties of plane polynomial automorphisms,Ergodic Theory Dyn. Syst.,9, 67–99, (1989). · Zbl 0651.58027 · doi:10.1017/S014338570000482X [13] Hubbard, J.H. Hénon mappings in the complex domain, inChaotic Dynamics and Fractals, Barnsley, M. and Demko, S., Eds., Academic Press, 101–111, (1986). [14] Hubbard, J.H. and Oberste-Vorth, R. Hénon mappings in the complex domain I: The global topology of dynamical space,Inst. Hautes Études Sci. Publ. Math.,79, 5–46, (1994). · Zbl 0839.54029 · doi:10.1007/BF02698886 [15] Ledrappier, F. and Young, L.-S. The metric entropy of diffeomorphisms, I & II,Ann. Math.,122, 509–539 and 540–574, (1985). · Zbl 0605.58028 · doi:10.2307/1971328 [16] 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 [17] Pollicott, M. Lectures on ergodic theory and Pesin theory on compact manifolds, London Mathematical Society Lecture Note Series 180, Cambridge University Press, (1993). · Zbl 0772.58001 [18] 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 [19] Pugh, C. and Shub, M. Ergodic attractors,Trans. AMS,312, 1–54, (1989). · doi:10.1090/S0002-9947-1989-0983869-1 [20] Sibony, N. Iteration of polynomials, U.C.L.A. Course Lecture Notes. [21] Tortrat, P.Aspects potentialistes de l’itération des polynômes, InSéminaire de Théorie du Potentiel Paris, No. 8, (Lect. Notes Math., 1235), Springer-Verlag, (1987). [22] Wu, H. Complex stable manifolds of holomorphic diffeomorphisms,Indiana U. Math. J.,42, 1349–1358, (1993). · Zbl 0811.58009 · doi:10.1512/iumj.1993.42.42062
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.