Upper bounds for quantum dynamics governed by Jacobi matrices with self-similar spectra

We study a class of one-sided Hamiltonian operators with spectral measures given by invariant and ergodic measures of dynamical systems of the interva l. We analyse dimensional properties of the spectral measures and prove upp er bounds for the asymptotic spread in time of wavepackets. These bounds in volve the Hausdorff dimension of the spectral measure, multiplied by a corr ection calculated from the dynamical entropy, the density of states, and th e capacity of the support. For Julia matrices, the correction disappears an d the growth is ruled by the fractal dimension.