×

On the dynamics near infinity of some polynomial mappings in \(\mathbb C^2\). (English) Zbl 1079.37040

Summary: We construct the Green current for a random iteration of horizontal-like mappings in \(\mathbb C^2\). This is applied to the study of a polynomial map \(f : \mathbb C^2 \to \mathbb C^2\) with the following properties: (i) infinity is \(f\)-attracting; (ii) \(f\) contracts the line at infinity to a point not in the indeterminacy set.
We study for such mappings the escape rates near infinity, i.e., the set of possible values of the function \(\lim\sup\frac1n\log^+\log^+\| f^n\|\). We show in particular that the set of possible values can contain an interval.
On the other hand, the Green current \(T\) of \(f\) can be decomposed into pieces associated to an itinerary defined by the indeterminacy points. This allows us to prove that \(\lim\frac1n\log^+\log^+\| f^n\|\) exists \(\| T\|\)-a.e. and we give its value in terms of explicit quantities depending on \(f\).

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
32U40 Currents
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Bedford, E., Jonsson, M.: Dynamics of regular polynomial endomorphisms of . Amer. J. Math. 122, 153–212 (2000) · Zbl 0941.37027
[2] Bedford, E., Smillie, J.: Real polynomial diffeomorphisms of maximal entropy: II. Small Jacobian. Preprint · Zbl 1135.37015
[3] Demailly, J.-P.: Monge-Ampère operators, Lelong numbers and intersection theory. Complex analysis and geometry, Univ. Ser. Math. Plenum, New York, 1993 pp 115–193 · Zbl 0792.32006
[4] Dinh, T.C., Sibony, N.: Dynamique des applications d’allure polynomiale. J. Math. Pures Appl. 82(9), 367–423 (2003) · Zbl 1033.37023
[5] Dinh, T.C., Sibony, N.: Dynamique des applications polynomiales semi-régulières. Ark. mat. 42, 61–85 (2004) · Zbl 1059.37033
[6] Dujardin, R.: Hénon-like mappings in . Amer. J. Math. 126, 439–472 (2004) · Zbl 1064.37035
[7] Duval, J., Sibony, N.: Polynomial convexity, rational convexity, and currents. Duke Math. J. 79, 487–513 (1995) · Zbl 0838.32006
[8] Fornæss, J.E., Sibony, N.: Hyperbolic maps on \(\mathbb{P}\)2. Math. Ann. 311, 305–333 (1998) · Zbl 0928.37006
[9] Gamelin, T.W.: Uniform algebras. Prentice-Hall, Inc., Englewood Cliffs, N. J. 1969 · Zbl 0213.40401
[10] Guedj, V.: Dynamics of polynomial mappings of . Amer. J. Math. 124, 75–106 (2002) · Zbl 1198.32007
[11] Hubbard, J.H., Oberste-Vorth, R.W.: Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials. Real and complex dynamical systems, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ. Dordrecht, 1995 pp 89–132 · Zbl 0874.54037
[12] Katok, Anatole, Hasselblatt, Boris,: Introduction to the modern theory of dynamical systems. Cambridge University Press, 1995 · Zbl 0878.58020
[13] Levenberg, N., Słodkowski, Z.: Pseudoconcave pluripolar sets in . Math. Ann. 312, 429–443 (1998) · Zbl 0919.32008
[14] Łojasiewicz, S.: Introduction to complex analytic geometry. Birkhäuser Verlag, Basel, 1991 · Zbl 0747.32001
[15] Pugh, C., Shub, M.: Ergodic attractors. Trans. Amer. Math. Soc. 312, 1–54 (1989) · Zbl 0684.58008
[16] Sibony, N.: Dynamique des applications rationnelles de \(\mathbb{P}\)k. Dynamique et géométrie complexes (Lyon, 1997), Panoramas et Synthèses, 8, 1999
[17] Sibony, N.: Quelques problèmes de prolongements de courants en analyse complexe. Duke Math. J. 52, 157–197 (1985) · Zbl 0578.32023
[18] Słodkowski, Z.: Uniqueness property for positive closed currents in . Indiana Univ. Math. J. 48, 635–652 (1999) · Zbl 0939.32006
[19] Yamagishi, Y.: On the local convergence of Newton’s method to a multiple root. J. Math. Soc. Japan 55, 897–908 (2003) · Zbl 1161.37330
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.