Abstract:

What dynamical features do two commuting functions share? It is know since Julia and Fatou that any two commuting rational functions have the same Julia set. Moreover, families of commuting rational functions have been fully classified by Ritt and Eremenko independently. Both questions are still unsolved for entire transcendental functions, due to the presence of wandering domains. We present an historical background and some little additional step towards the conjecture that also in this case, any two commuting functions have the same Julia set. This is a work in progress with Filippo Bracci.