  We show that the Mahler measures of the Jones polynomial and of the colored
Jones polynomials converge under twisting for any link. Moreover, almost all of
the roots of these polynomials approach the unit circle under twisting. In
terms of Mahler measure convergence, the Jones polynomial behaves like
hyperbolic volume under Dehn surgery. For pretzel links P(a_1,...,a_n), we show
that the Mahler measure of the Jones polynomial converges if all a_i tend to
infinity, and approaches infinity for a_i = constant if n tend to infinity,
just as hyperbolic volume. We also show that after sufficiently many twists,
the coefficient vector of the Jones polynomial and of any colored Jones
polynomial decomposes into fixed blocks according to the number of strands
twisted.
