  We study skew product systems driven by a hyperbolic base map S (e.g. a baker
map or an Anosov surface diffeomorphism) and with simple concave fibre maps on
an interval [0,a] like h(x)=g(\theta) tanh(x) where g(\theta) is a factor
driven by the base map. The fibre-wise attractor is the graph of an upper
semicontinuous function \phi(\theta). For many choices of the function g, \phi
has a residual set of zeros but \phi>0 almost everywhere w.r.t. the
Sinai-Ruelle-Bowen measure of S^(-1).
  In such situations we evaluate the stability index of the global attractor of
the system, which is the subgraph of \phi, at all regular points (\theta,0) in
terms of the local exponents \Gamma(\theta):=\lim_{n\to\infty} 1/n log
g_n(\theta) and \Lambda(\theta):=\lim_{n\to\infty} 1/n\log|D_u S^{-n}(\theta)|
and of the positive zero s_* of a certain thermodynamic pressure function
associated with S^(-1) and g. (In queuing theory, an analogon of s_* is known
as Loyne's exponent.)
  The stability index was introduced by Podvigina and Ashwin in 2011 to
quantify the local scaling of basins of attraction.
