  Limit-periodic structures are well ordered but nonperiodic, and hence have
nontrivial vibrational modes. We study a ball and spring model with a
limit-periodic pattern of spring stiffnesses and identify a set of extended
modes with arbitrarily low participation ratios, a situation that appears to be
unique to limit-periodic systems. The balls that oscillate with large amplitude
in these modes live on periodic nets with arbitrarily large lattice constants.
By studying periodic approximants to the limit-periodic structure, we present
numerical evidence for the existence of such modes, and we give a heuristic
explanation of their structure.
