Abstract
For any proper polynomial mapf: C k→C k define the function α as
Letf=(P 1,...,P k ) be a proper polynomial map. We define a notion ofs-regularity using the extension off to Pk. Whenf is (maximally) regular we show that the function α is lower semicontinuous and takes only finitely many values: 0 andd 1,...,d k , whered i :=degP i . We then describe dynamically the sets {α≤d i }. We give a concrete description of regular maps. Ifd i >1, this allows us to construct the equilibrium measure μ associated withf 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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Dinh, TC., Sibony, N. Dynamique des applications polynomiales semi-régulières. Ark. Mat. 42, 61–85 (2004). https://doi.org/10.1007/BF02385579
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DOI: https://doi.org/10.1007/BF02385579