Asymptotic Expansions for Fourier Transform of Singular Self-Affine Measures

Singular continuous self-affine measures invariant with respect to some stochastic dynamical systems generated by a finite set of affine contractions on the axis are studied. The Hausdorff dimension of their support and leading terms of L2-norm asymptotic of the corresponding cut-off characteristic function are computed. It is shown that the power-law behavior multiplied by some oscillating functions of a logarithm of a cut-off parameter for these terms is typical. The inverse reconstruction problem of the invariant measure from the high-frequency behavior of its characteristic function is discussed.