This talk concerns joint work with Xander Faber. Let K be a nonarchimedean local field of characteristic 0 and residue characteristic p>0. Let q=pf be the order of its residue field, and let CK be the completion of an algebraic closure of K. The group of continuous automorphisms Galc(CK/K) acts on the Berkovich Projective Line P1CK. We show that the Galois invariant locus in P1CK is a densely branched tree which properly contains the path-closure of P1(K), and is contained in a tube of path-distance radius 1/(p−1)∗[1+1/(p−1)] around the path-closure. The radius can probably be improved to 1/(p−1). The Galois invariant locus has q+1 branches at each type II point in the locus corresponding to a disc D(a,pb), with b rational, and no other branches. We construct a conjecturally dense subset of the Galois invariant locus. We also establish a conjecture of Benedetto, that each Galois invariant point is defined over a totally ramified extension of K.