  Two-dimensional layered aperiodic Ising systems are studied in the extreme
anisotropic limit where they correspond to quantum Ising chains in a transverse
field. The modulation of the couplings follows an aperiodic sequence generated
through substitution. According to Luck's criterion, such a perturbation
becomes marginal when the wandering exponent of the sequence vanishes. Three
marginal sequences are considered: the period-doubling, paper-folding and
three-folding sequences. They correspond to bulk perturbations for which the
critical temperature is shifted. The surface magnetization is obtained exactly
for the three sequences. The scaling dimensions of the local magnetization on
both surfaces vary continuously with the modulation factor. The low-energy
excitations of the quantum chains are found to scale as L^z with the size L of
the system. This is the behaviour expected for a strongly anisotropic system,
where z is the ratio of the exponents of the correlation lengths in the two
directions. The anisotropy exponent z is equal to the sum of the scaling
dimensions of the local magnetization on the two surfaces. The anisotropic
scaling behaviour is verified numerically for other surface and bulk critical
properties as well.
