Gibbs-non-Gibbs transitions for measures on Cayley trees

We consider Gibbs measures for Ising models on Cayley trees, subjected to an infinite-temperature Glauber dynamics (independent spin flips). We discuss similarities and differences with the situation on regular lattices. In particular we find that on trees there is the possibility of different Gibbs measures acquiring different Gibbsian properties, and the possibility of measures becoming "totally non-Gibbsian" in the sense that all spin configurations become points of discontinuity for some conditional probability. This is based on joint work with V.Ermolaev, G.Iacobelli and C.Kuelske