A technique often used in
topology involves the inductive modification of a given mapping in order to achieve a
limit mapping having certain prescribed properties. The following definition
will facilitate the discussion. Suppose X and Y are topological spaces, and
{W_{i}},i = 1,2,⋯ , is a countable collection of subsets of X. Then a sequence
{f_{i}},i ≧ 0, of mappings from X into Y is called stable relative to {W_{i}} if
f_{i}(X − W_{i}) = f_{i−1}(X − W_{ẋ}),i,= 1,2,⋯ . Note, in the above definition, that if
{W_{i}} is a locally finite collection, then lim_{i→∞}f_{i} is necessarily a well defined
mapping from X into Y , and is continuous if each f_{i} is continuous. In a typical
smoothing theorem, a C^{r}mapping f : M → N between C^{∞} differentiable manifolds
M and N is approximated by a C^{∞}mapping g : M → N, where the mapping g is
constructed as the limit of a suitable sequence {f_{i}}(with f_{0} = f) which is stable
relative to a locally finite collection {C_{i}} of compact subsets of M. On the other
hand, instead of improving f, it is also of interest to approximate f by a mapping g
which has bad behavior at, say, a dense set of points of M. In this paper, such a
mapping g is constructed as the limit of a sequence {f_{i}} (with f_{0} = f) which is
stable relative to {C_{i}}, but where the C_{i} are more “clustered” than a locally finite
collection. The case of interest here is where a sequence of homeomorphisms {H_{i}},
which is stable relative to {U_{i}}, necessarily converges to a homeomorphism.
Theorem 1 of this paper gives a sufficient condition that the latter be satisfied
for homeomorphisms of a metric space. In Theorem 1, the collection {U_{i}}
is not, in general, locally finite (in fact, the U_{i} satisfy a certain “nested”
condition). Theorem 1 is used to establish a result concerning the distribution of
homeomorphisms (of a differentiable manifold) which have a dense set of spiral
points.
