Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

ORBITS OF TURNING POINTS FOR MAPS OF FINITE GRAPHS AND INVERSE LIMIT SPACES

BRIAN RAINES

Abstract. In this paper we examine the topology of inverse limit spaces generated by maps of finite graphs. In particular we explore the way in which the structure of the orbits of the turning points affects the inverse limit. We show that if f has finitely many turning points each on a finite orbit then the inverse limit of f is determined by the number of elements in the ω-limit set of each turning point. We go on to identify the local structure of the inverse limit space at the points that correspond to points in the ω-limit set of f when the turning points of f are not necessarily on a finite orbit. This leads to a new result regarding inverse limits of maps of the interval.

1. Introduction

Every one-dimensional continuum is an inverse limit on finite graphs, and many, though not all, are homeomorphic to an inverse limit on a finite graph with a single bonding map. These spaces also naturally appear in dynamical systems. R.F. Williams showed that if a manifold diffeomorphism F has a one-dimensional hyperbolic attractor Λ (with associated stable manifold structure)then F restricted to Λ is topologically conjugate with the shift homeomorphism on an inverse limit of a piecewise monotone map f of some finite graph, [11], and Barge and Diamond, [2], remark that for any map f : G → G of a finite graph there is a homeomorphism F : ℝ^{3} → ℝ^{3} with an attractor on which F is conjugate to the shift homeomorphism on lim_{←}{G, f}. More recently, Anderson and Putnam, [1], have shown that the dynamics arising from a substitution tiling is often conjugate to the action of a shift-map on an inverse limit of a branched d-manifold. They then demonstrate how to use knowledge about the inverse limit space to compute the cohomology and K-theory of a space of tilings. Extending these ideas, Barge, Jacklitch and Vago, [4], use inverse limits induced by certain Markov maps on wedges of circles to analyze one-dimensional substitution tiling spaces and one-dimensional unstable manifolds of hyperbolic sets. Many of their results rely on showing that certain pairs of these inverse limit spaces are not homeomorphic.

2000 Mathematics Subject Classification. 54H20, 54F15, 37E25.

Key words and phrases. inverse limits, graph, continuum.

Brian Raines, Orbits of turning points for maps of finite graphs and inverse limit spaces, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 253–263, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204137.

It is often quite difficult to distinguish between inverse limit spaces, even when the dynamics of the bonding maps are very different. Many papers have been written to this end, [3], [5], [6], [8], and [10]. However most of the techniques have been focused on maps of the interval. Perhaps one of the easiest ways to decide if two inverse limit spaces are not homeomorphic is to count their endpoints. Barge and Martin have shown that the number of endpoints of lim_{←}{[0, 1], f} is the same as the number of elements in the ω-limit set of the turning points of the bonding map, f, when f has a dense orbit and finitely many turning points, [5].

In this paper we distinguish between these inverse limit spaces by showing that many of the points have neighborhoods homeomorphic to the product of a zero-dimensional set and (0, 1), and we show that the exceptional points are those that always project onto the ω-limit set of the turning points. We do this for inverse limits on graphs, but of course, the result holds for inverse limits on the interval. In the case of the interval, our theorem is still an extension of Barge and Martin’s result, because there are many bonding maps that give rise to a three-endpoint indecomposable continua that have more than three points in the ω-limit set of their turning points. Our theorem can be used to easily distinguish between these inverse limit spaces.

2. Preliminaries

By a continuum we mean a compact, connected, metric space, and by a mapping we mean a continuous function. We will say a mapping, f, is monotone on A if, and only if, f^{−1}(x) is connected for all x ∈ A. The inverse limit induced by a single bonding map, f, on a continuum M is defined as follows:

lim_{←}{M, f} = {(x_{0}, x_{1}, …)|x_{i} ∈ M and f(x_{i+1}) = x_{i}}.

Since M is metric and f is a mapping, lim_{←}{M, f} is a continuum with the metric:

d(x, y) = ∑_{i=0}^{∞} \frac{d_{M}(x_{i} − y_{i})}{2^{i}},

where d_{M} is the metric on M and we assume that d_{M}(x, y) < 1 for all x, y ∈ M. Define the projection maps π_{n} : lim_{←}{M, f} → M by π_{n}(x) = x_{n}, where x = (x_{1}, x_{2}, …) ∈ lim_{←}{M, f}. Also, define the shift homeomorphism

h : lim_{←}{M, f} → lim_{←}{M, f} by

h(x) = (f(x_{0}), f(x_{1}), f(x_{2}), …) = (f(x_{0}), x_{0}, x_{1}, …).

A linear chaining, or just chaining, of a continuum M is a finite sequence, L_{1}, L_{2}, L_{3}, …, L_{n} of open subsets of M such that L_{i} intersects L_{j} if and only if |i − j| < 2. The open sets comprising the chain are called the links of the chain. The mesh of a chain is the largest of the diameters of its links. A continuum M is said to be chainable provided that for each positive number ϵ there is a chaining of M with mesh less than ϵ, such a chain is called an ϵ-chain. It is a well-known fact that an inverse limit of chainable continua is a chainable continuum. A closed chain is a chain whose links are closed sets and if i ≠ j, then L_{i} ∩ L_{j} =Bd(L_{i})∩Bd(L_{j}) if, and only if, |i − j| < 2. We lose no generality in assuming that all of the chains in this paper are taut, i.e. if L_{i} ∩ L_{j} = ∅ then \overline{L_{i}} ∩ \overline{L_{j}} = ∅, [9]. Notice that if L is a taut chaining of an inverse limit space and L_{i} ∩ L_{j} = ∅ then it is possible to find a positive integer, q, large enough so that π_{q}(\overline{L_{i}})∩π_{q}(\overline{L_{j}}) = ∅, which implies that π_{q}(L) is a chain.

A finite graph, G, is a continuum that can be written as the union of finitely many arcs any two of which are either disjoint or intersect at only one of their endpoints. For any finite graph, G, there is a finite set of points called vertices, V = {v_{1}, v_{2}, …, v_{n}}, and a set of arcs, E, with endpoints from V called edges, with the property that if v_{k} ∈ e_{ij} ∈ E then either k = i or k = j, and if two edges meet, they meet only at a single common vertex. For simplicity, we adopt the convention that, since e_{ij} = e_{ji}, if we label an edge e_{ij} then i < j. For every point, x ∈ G, define the degree of x, deg(x), to be the number of edges in G that have x as an endpoint. Let V′ ⊆ V be the set of all points, x, with deg(x) ≥ 3.

Let a, b ∈ G. We will denote an arc between a and b by \overline{ab}. Clearly this arc is not uniquely determined. However if a, b ∈ e_{ij} then there is a unique arc with endpoints a and b that is contained in e_{ij}. We will denote this arc by [a, b], assuming that a is closer to v_{i} in the linear ordering of e_{ij} that has v_{i} as its least element, otherwise we denote it [b, a].

We will now extend the idea of linear-chains to graph-chains. Let n be a positive integer and let R be a relation on {1, 2 … n} with the property that if (i, j) ∈ R then i < j. For every (i, j) ∈ R, let C_{i,j} = {C_{1}^{i,j}, C_{2}^{i,j}, …, C_{n_{i,j}}^{i,j}} be a taut chain with the closure of every link of C_{i,j} being disjoint from the closure of every link of C_{k,l} whenever (k, l) ≠ (i, j), except C_{1}^{i,j} which meets every link of the form C_{1}^{i,k} and every C_{n_{m,i}}^{m,i} and also except for C_{n_{i,j}}^{i,j} which meets every link of the form C_{1}^{j,k} and every C_{n_{m,j}}^{m,j}. Call C_{i,j} an edge-chain. Let

C = ⋃_{(i,j)∈R} C_{i,j}.

Call C a graph-chain. Let G be a finite graph with vertex set, V, and edge set E. A graph-chaining of G is a graph-chain with R = {(i, j)|e_{ij} ∈ E} such that each vertex, v_{i}, is only in links of the form C_{1}^{i,j} or C_{n_{m,i}}^{m,i}, and each edge-chain, C_{i,j}, is a chaining of the corresponding edge, e_{ij}.

For a given graph-chain C, call the set E′ = R the edge index set. For notational convenience we will often denote a graph-chain by

C = {C_{1}^{i}, C_{2}^{i}, … C_{n_{i}}^{i}|i ∈ E′}

using i to represent an ordered pair in E′.

A continuum, M, is said to be graph-chainable provided that for each positive number ϵ there exists a graph-chaining of M with mesh less than ϵ. By a closed graph-chain we will mean a graph-chain C such that every link of C is closed and if A and B are different links in C with A ∩ B ≠ ∅, then A ∩ B =Bd(A)∩Bd(B). Notice that this implies that the only point in common to links of the form C_{1}^{i,j} and C_{n_{k,i}}^{k,i} is the vertex v_{i}.

It is easy to show that the inverse limit induced by maps on graphchainable continua is itself a graph-chainable continuum.

3. Markov Graph-Maps

First, we extend the definition of a Markov map of the interval (see [3] or [7])to a Markov map of a graph. Let f be a mapping of a finite graph, G, with vertex set V and edge set E and define a Markov graph-chaining, T^{f}, of G with respect to f to be a closed graph-chaining of G where, for every i ∈ E′, |T_{i}^{f}| = n_{i}, f restricted to each link is monotone but not constant, and for every i ∈ E′ and k ≤ n_{i} there exists a subset of E′ × ℕ, A_{i,k}, such that

f(T_{k}^{i}) = ⋃_{(p,r)∈A_{i,k}} T_{r}^{p}.

We will call a set of the form A_{i,k} the index set for (i, k) under f, and we will call a map that admits such a Markov graph-chain a Markov graph-map or simply a Markov map. The endpoints of each link of T^{f} determine a Markov partition of each edge. We define the Markov partition of the graph to be the set

B_{f} = {v_{i} = c_{0}^{i,j} < c_{1}^{i,j} < ⋯ < c_{n_{i,j}}^{i,j} = v_{j}|(i, j) ∈ E′}

where c_{k}^{i,j} and c_{k+1}^{i,j} are the endpoints of T_{k+1}^{i,j}. Notice that f(B_{f}) ⊆ B_{f}, f is not constant on [c_{k−1}^{i,j}, c_{k}^{i,j}], and f restricted to each such arc is monotone.

Also define the set S_{i,k} ⊂ E′ × ℕ such that (p, r) ∈ S_{i,k} if and only if [f^{−1}(T_{k}^{i})]^{∘} ∩ T_{r}^{p} ≠ ∅ i.e. S_{i,k} is the collection of indices of links of T_{f} that are mapped onto T_{k}^{i} by f. We will call the set S_{i,k} the inverse index set of (i, k) under f.

Let f : G_{1} → G_{2} be a a map between finite graphs G_{1} and G_{2}. Then x ∈ G_{1} is a turning point of f if there is an arc, \overline{ab} ⊆ G_{1}, containing x in its interior, such that f[\overline{ab}] = \overline{zf(x)} where z ∈ \{f(a), f(b)\} and both of f|_{A} and f|_{B} are monotone, where A = \overline{ax} ⊆ \overline{ab} and B = \overline{xb} ⊆ \overline{ab}. Denote the set of turning points of f by P_{f}.

Generally there is much freedom in determining the Markov chain; however we assume that the Markov chains used in this paper are “natural” in the sense that elements of the Markov partition are either vertices, turning points, or in the orbit of a turning point or vertex. In the next section we will need to assume that f^{−1}(x) consists of only isolated points, for all x ∈ G. The next theorem demonstrates that we lose no generality in assuming this when f is Markov.

Theorem 3.1. Let each of f and g be a Markov mapping of G, a finite graph, with associated Markov partitions, B_{f} = {c_{0}^{i} < c_{1}^{i} < ⋯ < c_{n_{i}}^{i}|i ∈ E′} and B_{g} = {d_{0}^{i} < d_{1}^{i} < ⋯ < d_{n_{i}}^{i}|i ∈ E′}. Suppose that for every i ∈ E′ and k ≤ n_{i} there is a p ∈ E′ and a r ≤ n_{p} such that f(c_{k}^{i}) = c_{r}^{p} if and only if g(d_{k}^{i}) = d_{r}^{p}, then lim_{←}{G, f} is homeomorphic to lim_{←}{G, g}.

Before presenting the proof of this theorem we will present a few useful facts about graph-chainable continua.

Let C be a closed graph-chaining of a continuum, M, with edge index set R, and let C′ be a refinement of C with edge index set R′. Let h be a function such that for every link, C_{k}^{′i} ∈ C′, let h(i, k) = (p, r) if and only if C_{k}^{′i} is a subset of C_{r}^{p} in C. In this case we shall say that C′ follows pattern h in C. The proof of the next theorem is quite obvious, and so it has been omitted.

Theorem 3.2. Let A be a graph-chainable continuum, and let {C_{i}}_{i=1}^{∞} be a sequence of refining graph-chainings of A such that

lim_{i→∞} mesh(C_{i}) = 0

and C_{i} follows pattern h_{i} in C_{i−1}. If B is also a graph-chainable continuum with a sequence of refining graph-chainings, {D_{i}}_{i=1}^{∞}, such that

lim_{i→∞} mesh(D_{i}) = 0

and D_{i} follows pattern h_{i} in D_{i−1} then A is homeomorphic to B.

Suppose that f : G → G is a Markov mapping of a finite graph, G, with vertex set V = {v_{1}, v_{2}, …, v_{n}} and edge set E, and let T^{f} be a Markov chain of G for f where, for every i ∈ E′, |T_{i}^{f}| = n_{i}. Suppose that C is a closed refinement of T^{f} with |C_{i}| = m_{i}, such that C follows pattern h in T^{f}. We define the Markov graph-chain function, f̂, on the elements of C by the following. (We denote the lexicographical ordering on E′ by ≪.)

First let j be the least integer, k, such that (1, k) ∈ E′ and define

f̂_{p,r}(C_{1}^{1,j}) = f^{−1}(C_{1}^{1,j}) ∩ T_{r}^{p}

where (p, r) ∈ S_{h((1,j),1)}. For (k, ℓ) ∈ E′ and (p, r) ∈ S_{h((k,ℓ),1)} let

f̂_{p,r}(C_{1}^{k,ℓ}) = [f^{−1}(C_{1}^{k,ℓ}) ∩ T_{r}^{p}] − [  ⋃_{(q,s)∈E′,(q,s)≪(k,ℓ)} f̂_{p,r}(C_{1}^{q,s})]^{∘} .

For (p, r) ∈ S_{h((1,j),n_{i,j})} define

f̂_{p,r}(C_{n_{1,j}}^{1,j}) = [f^{−1}(C_{n_{1,j}}^{1,j}) ∩ T_{r}^{p}] − [ ⋃_{(j,ℓ)∈E′} f̂_{p,r}(C_{1}^{j,ℓ})]^{∘} .

For (k, ℓ) ∈ E′ and (p, r) ∈ S_{h((k,ℓ),n_{k,ℓ})}, let

f̂_{p,r}(C_{n_{k,ℓ}}^{k,ℓ}) = f^{−1}(C_{n_{k,ℓ}}^{k,ℓ}) ∩ T_{r}^{p}

− ([ ⋃_{(ℓ,m)∈E′} f̂_{p,r}(C_{1}^{ℓ,m})]^{∘} ∪ [ ⋃_{(q,s)∈E′,(q,s)≪(k,ℓ)} f̂_{p,r}(C_{n_{q,s}}^{q,s})]^{∘}) .

Finally for any (j, k) ∈ E′, 1 < m < n_{j,k} and (p, r) ∈ S_{h((j,k),m)} let

\begin{aligned}f̂_{p,r}(C_{m}^{j,k}) & = & f^{−1}(C_{m}^{j,k}) ∩ T_{r}^{p} \\ & & − ([f̂_{p,r}(C_{m−1}^{j,k})]^{∘} ∪ [f̂_{p,r}(C_{1}^{j,k})]^{∘} ∪ [f̂_{p,r}(C_{n_{j,k}}^{j,k})]^{∘}) . \end{aligned}

Define

f̂(C) = {f̂_{p,r}(C_{k}^{i,j})|(i, j) ∈ E′, k ≤ n_{i} and (p, r) ∈ S_{h((i,j),k)}} .

Notice that since f restricted to each link of T^{f} is monotone, f restricted to each link of C is monotone. So each element of f̂(C) is connected.

Lemma 3.1. If C is a closed refinement of T^{f} then f̂(C) is a closed graphchain and the components of f̂(C) refine T^{f}.

Proof. For every i ∈ E′, let m_{i} = |C_{i}|. Every element of f̂(C) is closed and f̂(C) covers G. Suppose now that C_{k}^{i} ∩ C_{r}^{p} = ∅, but f̂(C_{k}^{i}) ∩ f̂(C_{r}^{p}) ≠ ∅. Let x ∈ f̂(C_{k}^{i}) ∩ f̂(C_{r}^{p}). This implies that f(x) ∈ C_{k}^{i} ∩ C_{r}^{p}, a contradiction. So the only elements of f̂(C) which intersect are images of links of C which intersected, and since C is a closed graph-chaining of G, f̂(C) is also a closed graph-chaining of G. By definition, links of f̂(C) intersect only on their boundary and the components of f̂(C) refine T^{f}. □

Now suppose that g is another Markov mapping of G and let

S^{g} = {S_{0}^{i}, … S_{n_{i}}^{i}|i ∈ E′}

be the Markov graph-chain associated with g where, for every i ∈ E′, |S_{i}| = n_{i}. Denote the corresponding Markov partition by

B_{g} = {d_{0}^{i,j} < d_{1}^{i,j} < ⋯ < d_{n_{i,j}}^{i,j}|(i, j) ∈ E′}.

We are now ready to prove Theorem 3.1.

Proof of 3.1. Choose a positive number δ_{1} such that if H is a closed graphchaining of G with mesh less than δ_{1} which refines T^{f} or S^{g} then π_{1}^{−1}(H) ∩ lim_{←}\{G, f\} and π_{1}^{−1}(H) ∩ lim_{←}\{G, g\} both have mesh less than \frac{1}{2}. For every i ∈ E′ and j ≤ n_{i} , let Q_{j}^{i} be a positive integer such that δ_{1} · Q_{j}^{i} > diam(T_{j}^{i}) and δ_{1} · Q_{j}^{i} > diam(S_{j}^{i}).

Let C_{1} be a closed graph-chaining of G with mesh less than δ_{1} which refines T^{f} such that, for every i ∈ E′ and j ≤ n_{i}, there are Q_{j}^{i} links of C_{1} contained in T_{j}^{i}. Let J_{1} be defined similarly with respect to g and S^{g}.

Let D_{1} = π_{1}^{−1}(C_{1}) ∩ lim_{←}{G, f} and let K_{1} = π_{1}^{−1}(J_{1}) ∩ lim_{←}{G, g}. It is

easy to see that both of D_{1} and K_{1} are closed graph-chainings of lim_{←}{G, f}

and lim_{←}\{G, g\} respectively with mesh less than \frac{1}{2}.

By lemma 3.1 both of f̂(C_{1}) and ĝ(J_{1}) are closed graph-chainings of G which refine T^{f} and S^{g} respectively. Let δ_{2} be a positive number so that any closed graph-chaining of G, H, with mesh less than δ_{2} has π_{2}^{−1}(H)∩lim_{←}{G, f}

and π_{2}^{−1}(H) ∩ lim_{←}\{G, g\} both have mesh less than \frac{1}{4}.

By the hypothesis of the theorem, f̂_{p,r}(C_{j}^{i}) is defined if and only if ĝ_{p,r}(J_{j}^{i}) is defined. Notice that by the construction of C_{1} and J_{1}, there is a function, ℓ, such that C_{1} follows pattern ℓ in T^{f} and J_{1} also follows pattern ℓ in S^{g}. So for every i ∈ E′, j ≤ n_{i}, and (p, r) ∈ S_{ℓ(i,j)}, let Q_{i,j}^{p,r} be a positive integer such that Q_{i,j}^{p,r} · δ_{2} > diam(f̂_{p,r}(C_{j}^{i})) and Q_{i,j}^{p,r} · δ_{2} > diam(ĝ_{p,r}(J_{j}^{i})). Let C_{2} be a refinement of f̂(C_{1}) such that there are Q_{i,j}^{p,r} links of C_{2} inside each f̂_{p,r}(C_{j}^{i}). Let J_{2} be a refinement of ĝ(J_{2}) defined similarly. Define D_{2} to be

π_{2}^{−1}(C_{2}) ∩ lim_{←}{G, f} and define K_{2} to be π_{2}^{−1}(J_{2}) ∩ lim_{←}{G, g}.

Notice that if A is a subset of f̂_{p,r}(C_{j}^{i}) then

π_{2}^{−1}(A) ∩ lim_{←}{G, f} ⊆ π_{1}^{−1}(C_{j}^{i}) ∩ lim_{←}{G, f},

and similarly if A is a subset of ĝ_{p,r}(J_{j}^{i}) then

π_{2}^{−1}(A) ∩ lim_{←}{G, g} ⊆ π_{1}^{−1}(J_{j}^{i}) ∩ lim_{←}{G, g}.

So, since we have exactly Q_{i,j}^{p,r} links of C_{2} and J_{2} in f̂_{p,r}(C_{j}^{i}) and ĝ_{p,r}(J_{j}^{i}) respectively, D_{2} follows the same pattern, h_{2}, in D_{1} that K_{2} follows in K_{1}.

Clearly, chains of lim_{←}{G, f} and lim_{←}{G, g}, D_{3} and K_{3} can be constructed

such that mesh(D_{3}) < \frac{1}{8}, mesh(K_{3}) < \frac{1}{8}, and both D_{3} and K_{3} follow pattern h_{3} in D_{2} and K_{2} respectively.

So it is easy to see that we can build a sequence of refining chainings, {D_{i}}_{i=1}^{∞}, of lim_{←}{G, f} such that D_{i} follows pattern h_{i} in D_{i−1} and

lim_{i→∞} mesh(D_{i}) = 0,

and we can build a sequence of refining chainings, {K_{i}}_{i=1}^{∞}, of lim_{←}{G, g} such that K_{i} follows pattern h_{i} in K_{i−1} and

lim_{i→∞} mesh(K_{i}) = 0.

Thus, by Theorem 3.2, lim_{←}{G, f} is homeomorphic to lim_{←}{G, g}. □

This theorem provides some justification for the assumption that we make in the next section that f^{−1}(x) is completely disconnected for all x ∈ G. It shows that for a bonding map with finitely many turning points each on a finite orbit that we lose no generality in making this assumption. It also has an interesting, and immediate, corollary which is an extension of a theorem of Holte ([7], Theorem 3.2).

Corollary 3.2.1. Let each of f and g be Markov maps of the interval with associated Markov partitions, B_{f} = {0 = c_{0} < c_{1} < ⋯ < c_{n} = 1}, and B_{g} = {0 = d_{0} < d_{1} < ⋯ < d_{n} = 1}. Suppose that f(c_{i}) = c_{j} if, and only if, g(d_{i}) = d_{j}, then lim_{←}{[0, 1, f} is homeomorphic to lim_{←}{[0, 1], g}.

This corollary extends Holte’s theorem in the sense that the Markov partitions for f and g do not need to be the same set of points in the interval. So it allows for not only eliminating flat spots, but also dramatically changing slopes and shifting turning points around.

4. Inverse Limit Spaces

In this section we will assume that f : G → G is continuous, has finitely many turning points and, for every x ∈ G, f^{−1}(x) is completely disconnected. We will also assume that for every y ∈ G there exists a positive integer n such that f^{−n}(y) consists of more than one point and if C ⊆ G is connected then diam(C′) ≤ diam(C) for every component, C′ of f^{−1}(C).

Let E_{G} ⊆ G be the set of endpoints of G. The ω-limit set of a point x under a mapping f, ω_{f}(x) or simply ω(x), is given by

ω(x) = ⋂_{N∈ℕ} \overline{\{f^{n}(x)|n ≥ N\}},

and the ω-limit set of a set, X, is given by ω(X) = ⋃_{x∈X} ω(x).

Denote the elements of P_{f}, the turning points of f, in the edge e_{i,j} by t_{1,0}^{i,j} < t_{2,0}^{i,j} ⋯ < t_{m,0}^{i,j}, where < is the linear ordering of the arc e_{i,j} that has v_{i} as its least element, and for every t_{k,0}^{i,j} denote the orbit of t_{k,0}^{i,j} by

orb(t_{k,0}^{i,j}) = {t_{k,ℓ}^{i,j} = f^{ℓ}(t_{k,0}^{i,j})|ℓ ∈ ℕ}.

Let

orb(P_{f}) = ⋃_{t_{k,0}^{i,j}∈P_{f}} orb(t_{k,0}^{i,j}).

Theorem 4.1. Let x ∈ lim_{←}{G, f} have the property that for some N ∈ ℕ, if n ≥ N then deg(x_{n}) ∈ {0, 2}. Then there is a positive number ϵ and a zero-dimensional set S with the property that B_{ϵ}(x) is homeomorphic to (0, 1) × S if, and only if:

(i) there is a positive number, δ, and a positive integer, s, such that B_{δ}(x_{s}) ∩ f^{p}(E_{G}) = ∅ for every p ≥ 0, and

(ii) there exists a positive integer, m, such that x_{m} ∉ ω(P_{f}).

Proof. Let N ∈ ℕ such that if n ≥ N then deg(x_{n}) < 3, and let m ≥ N be such that x_{m} ∉ ω(P_{f}). Since f[ω(P_{f})] ⊆ ω(P_{f}), if n ≥ m then x_{n} ∉ ω(P_{f}). Let n ≥ m. Since x_{n} ∉ ω(P_{f}), x_{n} ∉ ω(t_{k,0}^{i,j}) for every t_{k,0}^{i,j} ∈ P_{f}. By the definition of the ω-limit set, for every t_{k,0}^{i,j} ∈ P_{f}, there is a positive

integer, p_{k,0}^{i,j} with the property that x_{n} ∉ \overline{\{t_{k,r}^{i,j}|r ≥ p_{k,0}^{i,j}\}}. Since there are only finitely many turning points for f, there exists a positive integer, q, such that x_{n} ∉ \overline{\{t_{k,r}^{i,j}|r ≥ q\}} for all t_{k,0}^{i,j} ∈ P_{f}. So x_{n+q+1} is not in \overline{\{t_{k,r}^{i,j}|r ∈ ℕ\}} for all t_{k,0}^{i,j} ∈ P_{f}. Let p = n + q + 1. There is a positive number, λ_{p} < δ, such that

B_{λ_{p}}(x_{p}) ∩ [ ⋃_{t_{k,0}^{i,j}∈P_{f}}{t_{k,r}^{i,j}|r ∈ ℕ}] = ∅,

and if y ∈ B_{λ_{p}}(x_{p}) then deg(y) is either 0 or 2. So B_{λ_{p}}(x_{p}) is homeomorphic to (0, 1), and if q ∈ ℕ and A is any component of f^{−q}[B_{λ_{p}}(x_{p})] then A is homeomorphic to (0, 1). Let ϵ be a positive number such that π_{p}[B_{ϵ}(x)] ⊆ B_{λ_{p}}(x_{p}). Let A = π_{p}[B_{ϵ}(x)]. Let A_{1}, A_{2}, … A_{n} be the components of f^{−1}(A). For each i ≤ n, f|_{A_{i}} is monotone and A_{i} ∩ f^{m}(E) = ∅ for all m ≥ 0. For each i ≤ n, let A_{1}^{i}, A_{2}^{i}, … A_{n_{i}}^{i} be the components of f^{−1}(A_{i}). Again, for each j ≤ n_{i}, f|_{A_{j}^{i}} is monotone and A_{j}^{i} ∩ f^{m}(E) = ∅

for all m ≥ 0. Assuming that A_{k}^{i,j,...p,t} is a component of f^{−1}[A_{t}^{i,j,...p}], define A_{1}^{i,j,...k}, A_{2}^{i,j...k} … A_{n_{i,j,...k}}^{i,j,...k} to be the components of f^{−1}[A_{k}^{i,j,...t}].

Let S be a collection of sequences of positive integers such that ⟨y_{i}⟩ ∈ S if, and only if, A_{y_{1}} is defined and for every other i ∈ ℕ, A_{y_{i+1}}^{y_{1},...y_{i}} is defined. Clearly S is zero-dimensional. Each sequence, ⟨y_{i}⟩ in S defines a sequence of open arcs, A_{y_{1}}, A_{y_{2}}^{y_{1}}, …, with the property that f maps A_{y_{i+1}}^{y_{1},y_{2},...y_{i}} onto A_{y_{i}}^{y_{1},y_{2}...y_{i−1}} monotonically. So lim_{←}{A_{y_{i+1}}^{y_{1},y_{2},...y_{i}}, f} is homeomorphic to (0, 1), and

B_{ϵ}(x) = ⋃_{⟨y_{i}⟩∈S} lim_{←}{A_{y_{i+1}}^{y_{1},y_{2},...y_{i}}, f}

is homeomorphic to S × (0, 1).

Now assume that x does not satisfy either (i) or (ii) in the theorem. First let ϵ be a positive number, and consider the ϵ-neighborhood around x, B_{ϵ}(x). Let n be a positive integer and let γ be a positive number such that π_{n}^{−1}[B_{γ}(x_{n})] ⊆ B_{ϵ}(x), B_{γ}(x_{n}) meets P_{f} at at most one point, and if y ∈ B_{γ}(x_{n}) then deg(y) < 3.

If for every positive number δ < γ and every positive integer M, there exists a positive integer, m > M, such that every image under f^{−m} of B_{δ}(x_{n}) meets the set of endpoints for G, E_{G}, then clearly for every positive number, λ, we can λ-chain lim_{←}{G, f} with a linear subchain that starts at x. Thus there is a chainable endcontinuum in G having x as an endpoint, and B_{ϵ}(x) cannot be homeomorphic to (0, 1) × S where S is a zero-dimensional set.

Instead, now suppose that γ is small enough and n is large enough so that f^{−m}(B_{γ}(x_{n})) misses E_{G} for every positive integer m. Let A = B_{γ}(x_{n}), and as above, enumerate the components of the preimages of A, A_{1}, A_{2}, … A_{n}. Continuing as previously, let S be the set of sequences of positive integers, ⟨y_{i}⟩, where ⟨y_{i}⟩ ∈ S if, and only if, for every positive integer i, A_{y_{i+1}}^{y_{1},...y_{i}} is a component of f^{−1}(A_{y_{i}}^{y_{1},...y_{i−1}}). Let t_{k,0}^{i,j} ∈ P_{f} with x_{n} ∈ ω(t_{k,0}^{i,j}). Then for infinitely many positive integers, m, t_{k,m}^{i,j} ∈ A. So there are infinitely many connected subsets of G, A_{y_{r+1}}^{y_{1}...y_{r}} that contain t_{k,0}^{i,j}. If there exists one of these subsets, A_{y_{r+1}}^{y_{1},...y_{r}} that only meet B_{f} at the singleton t_{k,0}^{i,j} then, since these components do not contain any endpoints or vertices of G, every component of the preimages of A_{y_{r+1}}^{y_{1},...y_{r}} is mapped with a single fold across A_{y_{r+1}}^{y_{1},...y_{r}}. Thus B_{ϵ}(x) contains a subspace homeomorphic to a neighborhood of (0, 1) in the sin(1/x)-continuum, and it cannot be homeomorphic to (0, 1) × T, where T is a zero-dimensional set. If instead, for some A_{y_{r+1}}^{y_{1},...,y_{r}} we have A_{y_{r+1}}^{y_{1},...y_{r}} ∩ B_{f} is a finite set then clearly we can restrict the size of A_{y_{r+1}}^{y_{1},...,y_{r}} in order to make the subset meet B_{f} at a singleton and produce a similar subspace.

So suppose that each subset, A_{y_{r+1}}^{y_{1},...y_{r}} that contains a turning point meets B_{f} on an infinite set. Pick one of these subsets and call it A_{1}. Let A_{2} be a connected subset of G such that f^{n_{1}}(A_{2}) = A_{1} and A_{2} ∩ P_{f} ≠ ∅. Given A_{i} define A_{i+1} to be a connected subset of G with the property that f^{n_{i}}(A_{i+1}) = A_{i} and A_{i+1} meets P_{f}. Then since A is small enough to not meet P_{f} at two points and since all of these subsets are preimages of A, they must all meet P_{f} at a single point. Similarly, they must all miss V′ and the set of endpoints of G. Thus, for every positive integer, i, f|_{A_{i}} is a two-pass map, and f^{n_{i}}|_{A_{i+1}} is at least a two-pass map. Thus lim_{←}{A_{i}, f|_{A_{i}}} is an indecomposable subcontinuum, and B_{ϵ}(x) contains an indecomposable subcontinuum. Hence, B_{ϵ}(x) is not homeomorphic to (0, 1) × T where T is a zero-dimensional set. □

Corollary 4.1.1. Suppose that f and g are maps of [0, 1] with the properties listed above. Further suppose that |ω(P_{f})| = n and |ω(P_{g})| = m. If n ≠ m then lim_{←}{[0, 1], f} is not homeomorphic to lim_{←}{[0, 1], g}.

Proof. Notice that for every point in ω(P_{f}) there is a point in the inverse limit that either is an endpoint or is in a neighborhood homeomorphic to a neighborhood of (0, 1) in the sin(1/x)-continuum. Also notice that any other point in the inverse limit has a neighborhood homeomorphic to the product of (0, 1) with a zero-dimensional set, S. These properties are preserved by homeomorphism, and so any space homeomorphic to it must have the same number of points in the ω-limit set of its turning points. □

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Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom

E-mail address: raines@maths.ox.ac.uk