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In this paper we address the following question due to Marcy Barge: For what f : I → I is it the case that the inverse limit of I with single bonding map f can be embedded in the plane so that the shift homeomorphism f̂ extends to a diffeomorphism ([BB, Problem 1.5], [BK, Problem 3])? This question could also be phrased as follows: Given a map f : I → I, find a diffeomorphism F : R → R so that F restricted to its full attracting set, ⋂ k≥0 F (R), is topologically conjugate to f̂ : (I, f) → (I, f). In this situation, we say that the inverse limit space, (I, f), can be embedded as the full attracting set of F . The problem of realizing inverse limits of one-dimensional manifolds as attracting sets of diffeomorphisms was first addressed by R. F. Williams [W] and L. Block [Bl]. Williams constructed diffeomorphisms of S which, when restricted to indecomposable subsets of their non-wandering sets, are conjugate to shift maps on inverse limits of one-dimensional branched manifolds [W, Theorem C]. Block generalized this construction to self-maps of manifolds which satisfy the condition that all singularities lie in the stable manifolds of orbits of sinks [Bl, Theorem A]. In this paper, we consider maps of the interval which satisfy Block’s condition, and use a method similar to the one used by Block in [Bl] to embed inverse limits of the interval with this type of bonding map as attracting sets of diffeomorphisms of R. We obtain a conjugacy on the entire attracting set rather than just an indecomposable subset of the non-wandering set. Other related work includes [BM], [Ba], [M], and [Sz]. In [BM], M. Barge and J. Martin show that any inverse limit of the interval with a single bonding map can be embedded as the full attracting set of a homeomorphism of the plane. In [Ba], M. Barge gives conditions which guarantee that an

Embedding inverse limits of nearly Markov interval maps as attracting sets of planar diffeomorphisms