Sign-Singular Measures: Fast Magnetic Dynamos, and High-Reynolds-Number Fluid Turbulence

It is shown that signed measures (i.e., measures that take on both positive and negative values) may exhibit an extreme form of singularity in which oscillations in sign occur everywhere on arbitrarily fine scale. A cancellation exponent is introduced to characterize such measures quantitatively, and examples of significant physical situations which display this striking type of singular behavior are discussed.