Isometries of Kellendonk-Savinien spectral triples and Connes metrics

We study the large and the small isometry groups of Kellendonk-Savinien spectral triples associated to a choice function for a self-similar compact ultrametric Cantor set; in particular, we show that under reasonable assumptions they coincide. To characterize these isometry groups, we use the Michon rooted weighted tree associated to an ultrametric Cantor set: the small and large isometry groups turn out to be equal to the subgroup of the automorphism group of the tree consisting of elements that commute with the choice function in a suitable sense. When the rooted tree is binary, these isometry groups are equal to Z2. We also examine examples of Connes metrics associated to Pearson-Bellissard spectral triples, presenting a gamut of cases in where it is infinite.