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Let $f$ and $g$ be permutable transcendental entire functions. What can we say about properties that they may have in common?  This is a function theoretical question  that can be made dynamical by asking   whether $f$ and $g$ do need to have the same Julia set. I will overview the known results about this topic and present the remaining open cases.  This is joint work with P. Rippon and G. Stallard.