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April 2004 Dynamique des applications polynomiales semi-régulières
Tien-Cuong Dinh, Nessim Sibony
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Ark. Mat. 42(1): 61-85 (April 2004). DOI: 10.1007/BF02432910


For any proper polynomial map f: CkCk define the function α as $\alpha (z): = \mathop {\lim \sup }\limits_{n \to \infty } \frac{{\log ^ + \log ^ + \left| {f^n (z)} \right|}}{n},where log^ + : = \max \{ \log , 0\} .$ Let f=(P1,..., Pk) be a proper polynomial map. We define a notion of s-regularity using the extension of f to Pk. When f is (maximally) regular we show that the function α is lower semicontinuous and takes only finitely many values: 0 and d1,..., dk, where di:=degPi. We then describe dynamically the sets {α≤di}. We give a concrete description of regular maps. If di >1, this allows us to construct the equilibrium measure μ associated with f as a generalized intersection of positive currents. We then give an estimate of the Hausdorff dimension of μ. We extend the approach to the larger class of (π, s)-regular maps. This gives an understanding of the largest values of α. The results can be applied to construct dynamically interesting measures for automorphisms.


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Tien-Cuong Dinh. Nessim Sibony. "Dynamique des applications polynomiales semi-régulières." Ark. Mat. 42 (1) 61 - 85, April 2004.


Received: 15 November 2002; Revised: 6 October 2003; Published: April 2004
First available in Project Euclid: 31 January 2017

zbMATH: 1059.37033
MathSciNet: MR2056545
Digital Object Identifier: 10.1007/BF02432910

Rights: 2004 © Institut Mittag-Leffler


Vol.42 • No. 1 • April 2004
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