Abstract. Taking inverse limits of the one-parameter family of tent maps of the interval generates a one-parameter family of inverse limit spaces. We prove that, for a dense set of parameters, these spaces are locally, at most points, the product of a Cantor set and an arc. On the other hand, we show that there is a dense G δ set of parameters for which the corresponding space has the property that each neighborhood in the space contains homeomorphic copies of every inverse limit of a tent map.In 1967, R. F. Williams ([10]) proved that hyperbolic one-dimensional attractors are inverse limits of maps on branched one-manifolds. These attractors have the solenoid-like property of being everywhere locally homeomorphic with the product of a Cantor set and an arc. Also, for dissipation parameter near zero, most of the full attracting sets for maps in the Hénon family are homeomorphic with inverse limits of unimodal maps of the interval ([1]). Except at finitely many points (the points of a stable periodic orbit), these sets are locally homeomorphic with the product of a Cantor set and an arc (see the comment following Theorem 1).Computer-generated pictures, at first glance, suggest that other one-dimensional (but non-hyperbolic) attractors might have a similar local structure. In particular, the transitive Hénon attractors appear to be, at most points, locally the product of a Cantor set and an arc. However, 'blowing up' computer pictures of these attractors usually indicates the presence of 'hooks' in the midst of regions that, under less scrutiny, look like a Cantor set of nearly parallel arcs.In this paper we consider the local topological properties of a one-parameter family of conceptual models for the Hénon attractors, inverse limits of tent maps. We find the following: for a dense set of parameters, the inverse limit space is, except at finitely many points, the product of a Cantor set and an arc (Theorem 1). However, for a dense G δ set of parameters, the inverse limit space is nowhere locally homeomorphic with the product of a Cantor set and an arc. In this second case, the inverse limit spaces display a remarkable form of self-similarity and local recapitulation of the entire family: not only does every open set in each space contain a homeomorphic copy of the entire space, each open set also contains a homeomorphic copy of every other inverse limit space appearing in the tent family (Corollary 6). In a forthcoming paper, we prove that the set of parameters for which this holds has full measure.
