  The well known stability conjecture of Palis and Smale states that if a
diffeomorphism is structurally stable then the chain recurrent set is
hyperbolic. It is natural to ask if this type of results is true for an
individual chain class, that is, whether or not every structurally stable chain
class is hyperbolic. Regarding the notion of structural stability, there is a
subtle difference between the case of a whole system and the case of an
individual chain class. The later case is more delicate and contains additional
difficulties. In this paper we prove a result of this type for the later case,
with an additional assumption of codimension 1. Precisely, let $f$ be a
diffeomorphism of a closed manifold $M$ and $p$ be a hyperbolic periodic point
of $f$ of index 1 or $\dim M-1$. We prove if the chain class of $p$ is
structurally stable then it is hyperbolic. Since the chain class of $p$ is not
assumed in advance to be locally maximal, and since the counterpart of it for
the perturbation $g$ is defined not canonically but indirectly through the
continuation $p_g$ of $p$, the proof is quite delicate.
