Echoing and scaling in Einstein-Yang-Mills critical collapse

We confirm recent numerical results of echoing and mass scaling in the gravitational collapse of a spherical Yang-Mills field by constructing the critical solution and its perturbations as an eigenvalue problem. Because the field equations are not scale invariant, the Yang-Mills critical solution is asymptotically, rather than exactly, self-similar, but the methods for dealing with discrete self-similarity developed for the real scalar field can be generalized. We find an echoing period $\Delta =0.73784\mathrm{}\pm 0.00002$ and a critical exponent for the black hole mass $\gamma =0.1964\mathrm{}\pm 0.0007$.