Geodesic
Embedding inverse limits of nearly Markov interval maps as attracting sets of planar diffeomorphisms
Colloquium Mathematicum, Tome 68 (1995) no. 2, pp. 291-296

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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 $\widehat f$ 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 $F:ℝ^2 → ℝ^2$ so that F restricted to its full attracting set, $⋂_{k ≥ 0} F^k(ℝ^2)$, is topologically conjugate to $\widehat f:(I,f) → (I,f)$. In this situation, we say that the inverse limit space, (I,f), can be embedded as the full attracting set of F. 
Sarah Holte. Embedding inverse limits of nearly Markov interval maps as attracting sets of planar diffeomorphisms. Colloquium Mathematicum, Tome 68 (1995) no. 2, pp. 291-296. doi: 10.4064/cm-68-2-291-296
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