Dynamics of commuting rational maps on Berkovich projective space over ℂ.

We study the dynamics of commuting rational maps with coefficients in ℂ. By lifting the dynamics from ℙ(ℂ) to Berkovich projective space ℙ, we prove that two nonlinear commuting maps have the same Berkovich Julia set and the same canonical measure. As a consequence, two nonlinear commuting maps with coefficient in ℂ have the same classical Julia set. We also prove that they have the same pre-periodic Berkovich Fatou components. [ABSTRACT FROM AUTHOR]