Fractal failure of quasilocality for a majority rule transformation on a tree (Q5943721)

Let \(\mu \) be a Gibbs measure of the Ising model on the rooted Cayley tree \(T^2_0\) and \(\nu \) its image under the ``majority rule'' transformation \(T:\{ -1,+1\} ^{T^2_0}\to \{ -1,0,+1\} ^{T^2_0}\) defined by \(T(\omega)_j=\omega _j\) iff \(\omega _{j0}=\omega _{j1}=\omega _j\) (where \(j0,j1\) are the two successors of \(j\)), and \(T(\omega)_j=0\), otherwise. It is shown that \(\nu \) is not quasilocal, and even more, that the ''nonquasilocality is fractal'', meaning that the Hausdorff dimension of the configurations leading to an essential discontinuity is positive.