An attractor \(A\) for a dynamical system is a set towards which a large number of orbits evolves under iteration. The basin of attraction \(B(A)\) is the set of initial conditions which, under iteration, converge to \(A\). Many dynamical systems have two or more attractors and the basins of attraction may have very complicated local geometry. The stability index (introduced by Podvigina-Ashwin and Greborgi-McDonald-Ott-Yorke) is a quantitative measure of the complexity of this local geometry and it behaves like a local dimension. We study the stability index for a family of skew-products in which the boundary between two attractors is a generalisation of Weierstrass' classical example of a nowhere differentiable function. We'll show how to calculate the stability index via techniques from thermodynamic formalism. This is joint work with Tom Withers.