Details

Thomas Gauthier ^[a] and Gabriel Vigny ^[b]

Distribution of points with prescribed derivative in polynomial dynamics

Pages: 247-270
Received: 24 June 2016
Accepted in revised form: 6 June 2017
Mathematics Subject Classification (2010): 37F10, 37F45, 32Uxx.
Keywords: Polynomial dynamics, Value distribution of derivatives, Equilibrium measure, Bifurcation current.
Author address:
[a],[b]: University of Picardie Jules Verne, 33 rue Saint-Leu, Amiens, 80039, France

Abstract: In analogy to the equidistribution of preimages of a prescribed point by the iterates of a polynomial map /(f/) in /(\C/) towards the equilibrium measure, we show here the equidistribution of points /(z/) for which /((f^n)'(z)=a/) for suitable /(a/) towards the equilibrium measure. We then give a similar statement in the space of degree /(d/) polynomials for the equidistribution of parameters for which the /(n/)-derivative at a given critical value has a prescribed derivative towards the activity current of the corresponding critical point.

This research was partially supported by the ANR grant Lambda ANR-13-BS01-0002. The first author is partially supported by a PEPS "Jeune-s Chercheur-e-s" Grant.

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