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.

众所周知，爱因斯坦-轴距-狄拉通系统中的引力临界坍塌会导致以Choptuik临界指数\（\ gamma \）为特征的连续自相似解。我们通过计算线性扰动方程来完善关于该主题的现有文献，其中轴力-狄拉通系统采用抛物线形式。接下来，在双曲情形下，我们在新发现的自相似解中求解摄动方程，这使我们能够提取Choptuik指数。我们的主要结果是，该指数不仅取决于时空的大小，而且还取决于特定的ansatz以及从其开始的关键解决方案。