  Gravitational critical collapse in the Einstein-axion-dilaton system is known
to lead to continuous self-similar solutions characterized by the Choptuik
critical exponent $\gamma$. We complete the existing literature on the subject
by computing the linear perturbation equations in the case where the
axion-dilaton system assumes a parabolic form. Next, we solve the perturbation
equations in a newly discovered self-similar solution in the hyperbolic case,
which allows us to extract the Choptuik exponent. Our main result is that this
exponent depends not only on the dimensions of spacetime but also the
particular ansatz and the critical solutions that one started with.
