  Let (U \subset {\mathbb R}^3) be an open set and (f:U \to f(U) \subset
{\mathbb R}^3) be a homeomorphism. Let (p \in U) be a fixed point. It is known
that, if (\{p\}) is not an isolated invariant set, the sequence of the fixed
point indices of the iterates of (f) at (p), ((i(f^n,p))_{n\geq 1}), is, in
general, unbounded. The main goal of this paper is to show that when (\{p\}) is
an isolated invariant set, the sequence ((i(f^n,p))_{n\geq 1}) is periodic.
Conversely, we show that for any periodic sequence of integers ((I_n)_{n
\geq1}) satisfying Dold's necessary congruences, there exists an orientation
preserving homeomorphism such that (i(f^n,p)=I_n) for every (n\geq 1). Finally
we also present an application to the study of the local structure of the
stable/unstable sets at (p).
