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

We study the dynamics of commuting rational maps with coefficients in . p . By lifting the dynamics from .1(. p ) to Berkovich projective space . 1Berk, 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 . p have the same classical Julia set. We also prove that they have the same pre-periodic Berkovich Fatou components.