  We consider semi-infinite two-dimensional layered Ising models in the extreme
anisotropic limit with an aperiodic modulation of the couplings. Using
substitution rules to generate the aperiodic sequences, we derive functional
equations for the surface magnetization. These equations are solved by
iteration and the surface magnetic exponent can be determined exactly. The
method is applied to three specific aperiodic sequences, which represent
different types of perturbation, according to a relevance-irrelevance
criterion. On the Thue-Morse lattice, for which the modulation is an irrelevant
perturbation, the surface magnetization vanishes with a square root
singularity, like in the homogeneous lattice. For the period-doubling sequence,
the perturbation is marginal and the surface magnetic exponent varies
continuously with the modulation amplitude. Finally, the Rudin-Shapiro
sequence, which corresponds to the relevant case, displays an anomalous surface
critical behavior which is analyzed via scaling considerations: Depending on
the value of the modulation, the surface magnetization either vanishes with an
essential singularity or remains finite at the bulk critical point, i.e., the
surface phase transition is of first order.
