Sala Conferenze (Puteano, Centro De Giorgi).
Let f and g be permutable transcendental entire functions. It is a long standing question whether the Julia sets of f and g are equal. The conjecture is true in the case that f,g are rational functions, and in the transcendental case the main issues are due to the presence of Baker Domains and Wandering Domains. Using a recent analysis of the dynamical behaviour in multiply connected wandering domains, we can make progress by showing that J(f) = J(g) unless (possibly) either f or g has a special type of wandering domains, namely simply connected wandering domain in the fast escaping set. This is joint work with G. Stallard and P. Rippon.