Embeddability into the plane and movability on inverse limits of graphs whose shift maps are expansive

AbstractIn [20, 21], Plykin proved that for each n=3, 4,… there exists a map f of a graph G onto itself such that (i) n=rank H1(G), (ii) the shift map f̃ of f is expansive and (iii) the inverse limit (G,f) of f can be embedded into the plane R2 and R2–(G,f) has (n+1)-components. In this paper, we prove that if G is any graph with rank H1(G)⩽2, there is no map of f of G onto itself such that the shift map f̃ is expansive and the inverse limit (G,f) can be embedded into the plane R2. This implies that Pykin's example is the best possible. Also, we prove the following results: (1) There is a map f of a graph G onto itself such that rank H1(G)=2, the shift map f̃ is expansive and the inverse limit (G,f) is movable. (2) If G is any graph with rank H1(G)=1, there is no map f of G onto itself such that the shift map f̃ is expansive and the inverse limit (G,f) is movable. It is known that for any graph G, there is a map f of G onto itself such that the shift map f̃ is expansive if and only if rank H1(G)⩾1 [11]