Inverse limits of certain interval mappings as attractors in two dimensions

Witold Szczechla

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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 $\hat{f}$ 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 $F : ℝ^{2} \to ℝ^{2}$ so that F restricted to its full attracting set, $⋂_{k \geq 0} F^{k} (ℝ^{2})$ , is topologically conjugate to $\hat{f} : (I, f) \to (I, f)$ . In this situation, we say that...