Aperiodic extended surface perturbations in the Ising model

We study the influence of an aperiodic extended surface perturbation on the surface critical behaviour of the two-dimensional Ising model in the extre me anisotropic limit. The perturbation decays as a power kappa of the dista nce l from the free surface with an oscillating amplitude A(l) = (-1)(fl)A where f(l) = 0, 1 follows some aperiodic sequence with an asymptotic densit y equal to 1/2 so that the mean amplitude vanishes. The relevance of the pe rturbation is discussed by combining scaling arguments of Cordery and Burkh ardt for the Hilhorst-van Leeuwen model and Luck for aperiodic perturbation s. The relevance-irrelevance criterion involves the decay exponent kappa, t he wandering exponent omega which governs the fluctuation of the sequence a nd the bulk correlation length exponent nu. Analytical results are obtained for the surface magnetization which displays a rich variety of critical be haviours in the (kappa, omega)-plane. The results are checked through a num erical finite-size-scaling study. They show that second-order effects must be taken into account in the discussion of the relevance-irrelevance criter ion. The scaling behaviours of the first gap and the surface energy are als o discussed.