DISTRIBUTION OF POSTCRITICALLY FINITE POLYNOMIALS II: SPEED OF CONVERGENCE

作者：Gauthier Thomas^*; Vigny Gabriel

来源：Journal of Modern Dynamics, 2017, 11(03): 57-98.

DOI：10.3934/jmd.2017004

• Bifurcation measure
• polynomial dynamics
• equidistribution
• hyperbolic polynomials
• speed estimate
• postcritically finite polynomials

In the moduli space of degree d polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for C-2-observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family and Favre and the first author in the space of degree d polynomials.
We deduce from that the equidistribution of hyperbolic parameters with (d - 1) distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for C-1-observables. As an application, we prove the equidistribution (up to an explicit extraction) of parameters with (d - 1) distinct cycles with prescribed multiplier toward the bifurcation measure for any (d - 1) multipliers outside a pluripolar set.

• 出版日期2017