Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 563-574

VALUE DISTRIBUTION OF THE SEQUENCES OF THE DERIVATIVES OF ITERATED POLYNOMIALS

Yusuke Okuyama

Kyoto Institute of Technology, Division of Mathematics
Sakyo-ku, Kyoto 606-8585, Japan; okuyama 'at' kit.ac.jp

Abstract. We establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any value in C \ {0} under the derivatives of the iterations of a polynomials f ∈ C[z] of degree more than one towards the f-equilibrium (or canonical) measure μ_f on P¹. We also show that for every C² test function on P¹, the convergence is exponentially fast up to a polar subset of exceptional values in C. A parameter space analog of the latter quantitative result for the monic and centered unicritical polynomials family is also established.

2010 Mathematics Subject Classification: Primary 37F10; Secondary 30D35, 32H50.

Key words: Value distribution, equidistribution, quantitative equidistribution, derivative, iterated polynomials, monic and centered unicritical polynomials family, complex dynamics, Nevanlinna theory.

Reference to this article: Y. Okuyama: Value distribution of the sequences of the derivatives of iterated polynomials. Ann. Acad. Sci. Fenn. Math. 42 (2017), 563-574.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4233