Extending Maps of a Cantor Set Product with an Arc to near Homeomorphisms of the 2-disk

In 1990, M. Barge and J. Martin [BM90] proved that the shift map on the inverse limit space ([0, 1], f), for any map f : [0, 1]→ [0, 1], can be realized as a global attractor in the plane. In 1960, M. Brown [Bro60] proved that the inverse limit space of any near homeomorphism (Definition 1.2) of a compact metric space is homeomorphic to the original space. M. Barge and J. Martin prove that, for all such f , there exists an embedding h : [0, 1]→ D2 such that h◦f ◦h−1 can be extended to a near homeomorphism of the 2-disk, D2. They then use M. Brown’s theorem to extend the induced shift homeomorphism on h([0, 1]) to a homeomorphism of D2. With care in the construction of the near homeomorphism of D2, the inverse limit space (h([0, 1]), h◦f ◦h−1) becomes a global attractor. The main goal of this paper is to show that analogous techniques for maps, F : C×[0, 1]→ C×[0, 1], where C is a Cantor set, F (x, y) = (F1(x), F2(x, y)) is a surjective map with positive topological entropy (Definition 1.4), and F1 is a homeomorphism, do not work; no near homeomorphic extension of h ◦ F ◦ h−1 to D2 exists for any embedding h : C × [0, 1] → D2 (Theorem 3.1). In our terminology, such F cannot be “embedded” into any 2-disk homeomorphism (Definition 1.1). In the proof of Theorem 3.1 one first assumes that h is a “tame” embedding (Definition 3.1). But in recent work, R. Walker proves that all embeddings of C × [0, 1] into D2 are tame [Wal]. Our study of maps of C×[0, 1] and their embeddings has links to a central problem in the dynamical systems of positive entropy homeomorphisms of compact surfaces.