On isotopy and unimodal inverse limit spaces

We prove that every self-homeomorphism $h : K_s \to K_s$ on the inverse limit space $K_s$ of tent map $T_s$ with slope $s \in (\sqrt 2, 2]$ is isotopic to a power of the shift-homeomorphism $\sigma^R : K_s \to K_s$.


Introduction
The solution of Ingram's Conjecture constitutes a major advancement in the classification of unimodal inverse limit spaces and the group of self-homeomorphisms on them. The first results towards solving this conjecture were obtained for tent maps with a finite critical orbit [9,12,3]. Raines andŠtimac [11] extended these results to tent maps with a possibly infinite, but non-recurrent critical orbit. Recently Ingram's Conjecture was solved completely (in the affirmative) in [2], but we still know very little of the structure of inverse limit spaces (and their subcontinua) for the case that orb(c) is infinite and recurrent, see [1,5,8].
Given a continuum K and x ∈ K, the composant A of x is the union of the proper subcontinua of K containing x. For slopes s ∈ ( √ 2, 2], the core is indecomposable (i.e., it cannot be written as the union of two proper subcontinua), and in this case we also proved [2] that any selfhomeomorphism h : K s → K s is pseudo-isotopic to a power σ R of the shift-homeomorphism σ on the core. This means that h permutes the composants of the core of K s in the same way as σ R does, and it is a priori a weaker property than isotopy. This is for instance illustrated by the sin 1 x -continuum, defined as the graph {(x, sin 1 x ) : x ∈ (0, 1]} compactified with a bar {0} × [−1, 1]. There are homeomorphisms that reverse the orientation of the bar, and these are always pseudo-isotopic, but never isotopic, to the identity. Since such sin 1 x -continua are precisely the non-trivial subcontinua of Fibonacci-like inverse limit spaces [8], this example is very relevant to our paper.
In this paper we make the step from pseudo-isotopy to isotopy. To this end, we exploit socalled folding points, i.e., points in the core of K s where the local structure of the core of K s is not that of a Cantor set cross an arc. In the next section we prove the following results: Folding points x = (. . . , x −2 , x −1 , x 0 ) are characterized by the fact that each entry x −k belongs to the omega-limit set ω(c) of the turning point c = 1 2 , see [10]. This gives the immediate corollary for those slopes such that the critical orbit orb(c) is dense in [c 2 , c 1 ], which according to [7]  The paper is organized as follows. In Section 2 we give basic definitions and prove results on how homeomorphisms act on folding points, i.e., Theorem 1.1 and Corollary 1.2. These proofs depend largely on the results obtained in [2]. In Section 3 we present the additional arguments needed for the isotopy result and finally prove Theorem 1.3. . Also let orb(c) and ω(c) be the orbit and the omega-limit set of c. We will restrict T s to the interval I = [0, s/2]; this is larger than the core [c 2 , c 1 ] = [s − s 2 /2, s/2], but it contains the fixed point 0 on which the 0-composant C 0 is based. , T s ) of0 will be denoted as C 0 ; it is a ray converging to, but disjoint from the core lim ← − ([c 2 , c 1 ], T s ) of the inverse limit space. We fix s ∈ ( √ 2, 2]; for these parameters T s is not renormalizable and lim ← − ([c 2 , c 1 ], T s ) is indecomposable. Moreover, the arc-component of0 coincides with the composant of0, but for points in the core of K s , we have to make the distinction between arc-component and composant more carefully.
In particular, x 0 = T p+l s (c). By convention, the endpoint 0 of C 0 is also a p-point and L p (0) := ∞, for every p. The ordered set of all p-points of the composant C 0 is denoted by E p , and the ordered set of all p-points of p-level l by E p,l . Given an arc A ⊂ K s with successive p-points x 0 , . . . , x n , the p-folding pattern of A is the sequence Note that every arc of C 0 has only finitely many p-points, but an arc A of the core of K s can have infinitely many p-points. In this case, if (u i ) i∈Z is the sequence of successive p-points of A, then F P p (A) = (L p (u i )) i∈Z . The folding pattern of the composant C 0 , denoted by F P (C 0 ), is the sequence L p (z 1 ), L p (z 2 ), . . . , L p (z n ), . . . , where E p = {z 1 , z 2 , . . . , z n , . . . } and p is any nonnegative integer. Let q ∈ N, q > p, and E q = {y 0 , y 1 , y 2 , . . . }. Since σ q−p is an orderpreserving homeomorphism of C 0 , it is easy to see that σ q−p (z i ) = y i for every i ∈ N, and L p (z i ) = L q (y i ). Therefore, the folding pattern of C 0 does not depend on p.
We call p-points satisfying this property salient.
Since for every slope s > 1 and p ∈ N 0 , the folding pattern of the 0-composant C 0 starts as ∞ 0 1 0 2 0 1 . . . , and since by definition L p (s 1 ) > 0, we have L p (s 1 ) = 1. Also, since Note that the salient p-points depend on p: if p ≥ q, then the salient p-point s i equals the salient q-point s i+p−q .
A folding point is any point x in the core of K s such that no neighborhood of x in core of K s is homeomorphic to the product of a Cantor set and an arc. In [10] it was shown that We can characterize folding points in terms of p-points as follows: Proof. ⇒ Take m ≥ p arbitrary. Since π m (x) ∈ ω(c) there is a sequence of post-critical points c n i → π m (x). This means that any point Since m is arbitrary, we can construct a diagonal sequence (x k ) k∈N of p-points, by taking a single element from (y i ) i∈N for each m, such that sup j≤k |π j ( We are now ready to prove Theorem 1.1. Proof of Corollary 1.2. If orb(c) is dense in [c 2 , c 1 ], every point x in the core of K s satisfies π k (x) ∈ ω(c) for all k ∈ N. By [10], this means that every point is a folding point, and hence the previous theorem implies that h ≡ σ R on the core of K s .

Remark 2.4.
A point x ∈ K s is an endpoint of an atriodic continuum, if for every pair of subcontinua A and B containing x, either A ⊂ B or B ⊂ A. The notion of folding point is more general than that of end-point. An example of a folding point that is not an endpoint is the midpoint x of a double spiral S, i.e., a continuous image of R containing a single folding point x and two sequences of p-points . . . y k ≺ y k+1 ≺ · · · ≺ x ≺ · · · ≺ z k+1 ≺ z k . . .
converging to x such that the arc-lengthd(y k , y k+1 ),d(z k , z k+1 ) → 0 as k → ∞. Here ≺ denotes the induced order on S.
It is natural to classify arc-components A according to the folding points they may contain. For arc-components A, we have the following possibilities: • A contains no folding point.
• A contains one folding point x, e.g. if x is an end-point of A or A is a double spiral.
• A contains two folding points, e.g. if A is the bar of a sin 1 x -continuum. • A contains countably many folding points. One can construct tent maps such that the folding points of its inverse limit space belong to finitely many arc-components that are periodic under σ, but where there are still countably folding points. 1 • A contains uncountably many folding points, e.g. if ω(c) = [c 2 , c 1 ], because then every point in the core is a folding point. This is clearly only a first step towards a complete classification.
Definition 2.5. Let ℓ 0 , ℓ 1 , . . . , ℓ k be those links in C p that are successively visited by an arc be the corresponding arc components such that Cl A i are subarcs of A. We call the arc A • p-link symmetric if ℓ i = ℓ k−i for i = 0, . . . , k; • maximal p-link symmetric if it is p-link symmetric and there is no p-link symmetric arc B ⊃ A and passing through more links than A.
The p-point of A k/2 with the highest p-level is called the center of A, and the link ℓ k/2 is called the central link of A.

Isotopic Homeomorphisms of Unimodal Inverse Limits
It is shown in [2] that every salient p-point s l ∈ C 0 is the center of the maximal p-link symmetric arc A l . We denote the central link that s l belongs to by ℓ s l p . For a better understanding of this section, let us mention that a key idea in [2] is that under a homeomorphism h such that h(C q ) ≺ C p , (maximal) q-link symmetric arcs have to map to (maximal) p-link symmetric arcs, and for this reason h(s m ) ∈ ℓ s l p for some appropriate m ∈ N (see [2, Theorem 4.1]).
Lemma 3.1. Let h : K s → K s be a homeomorphism pseudo-isotopic to σ R , and let q, p ∈ N 0 be such that h(C q ) C p . Let x be a q-point in the core of K s and let ℓ s l p ∈ C p be the link 1 An example is the tent-map where c 1 has symbolic itinerary (kneading sequence) ν = 100101 2 01 3 01 4 01 5 . . . . Then the two-sided itineraries of folding points are limits of {σ j (ν)} j≥0 . The only such two-sided limit sequences are 1 ∞ .1 ∞ and {σ j (1 ∞ .01 ∞ ) : j ∈ Z}. Since they all have left tail . . . 1111, these folding points belong to the arc-component of the point (. . . , p, p, p) for the fixed point p = s 1+s . This use of two-sided symbolic itineraries was introduced for inverse limit spaces in [6].
containing both σ p (x) and salient p-point s l , where l = L p (σ R (x)). Suppose that the arccomponent W x of ℓ s l p containing σ R (x) does not contain any folding point. Then h(x) ∈ W x .
Proof. Since W x does not contain any folding point, it contains finitely many p-points. Note that W x contains at least one p-point since σ R (x) ∈ W x is a p-point. Since C 0 is dense in K s , there exists a sequence (W i ) i∈N of arc-components of ℓ s l p such that W i ⊂ C 0 , F P p (W i ) = F P p (W x ) for every i ∈ N, and W i → W x in the Hausdorff metric. Let (x i ) i∈N be a sequence of q-points such that for every i ∈ N, It follows by the construction in the proof of [2,Proposition 4 Corollary 3.2. Let h : K s → K s be a homeomorphism pseudo-isotopic to σ R . Then h permutes arc-components of K s in the same way as σ R .
Proof. Since h is a homeomorphism, h maps arc-components to arc-components. Let A be an arc-component of K s . Let us suppose that A contains a folding point, say x. Then Let us assume now that A does not contain any folding point. There exist q, p ∈ N 0 such that h(C q ) C p and that h(A) is not contained in a single link of C p . Then A is not contained in a single link of C q . Let ℓ q ∈ C q and V ∈ ℓ q ∩ A be an arc-component of ℓ q such that V contains at least one q-point, say x. Let ℓ s l p ∈ C p be such that l = L p (σ R (x)). Let W ⊂ ℓ s l p be arc-component containing σ R (x). Since  Proof. Let us first suppose that h : K s → K s is any homeomorphism. Then, by [2, Theorem 1.2] there is an R ∈ Z such that h, restricted to the core, is pseudo-isotopic to σ R , i.e., h permutes the composants of the core of the inverse limit in the same way as σ R . Therefore, by Corollary 3.2, it permutes the arc-components of the inverse limit in the same way as σ R .
Let A, A ′ be arc-components of the core such that h, σ R : A → A ′ , and let x, y ∈ A, x ≺ y. We want to prove that h(x) ≺ h(y) if and only if σ R (x) ≺ σ R (y). Since h and σ R are homeomorphisms on arc-components, each of them could be either order preserving or order reversing. Therefore, to prove the claim we only need to pick two convenient points u, v ∈ A, u ≺ v, and check if we have either h(u) ≺ h(v) and σ R (u) ≺ σ R (v), or h(v) ≺ h(u) and σ R (v) ≺ σ R (u). If A contains at least two folding points, we can choose u, v to be folding points. Then h(u) = σ R (u) and h(v) = σ R (v) and the claim follows.
Let us suppose now that A contains at most one folding point. Then there exist q, p ∈ N 0 such that h(C q ) C p and q-points u, v ∈ A, u ≺ v (on the same side of the folding point if there exists one) such that σ R (u) and σ R (v) are contained in disjoint links of C p each of which does not contain the folding point of A, if there exists one.
Let ℓ s j p , ℓ s k p ∈ C p with j = L p (σ R (u)) and k = L p (σ R (v)) be links containing σ R (u) and σ R (v) respectively. Let W u ⊂ ℓ s j p and W v ⊂ ℓ s k p be arc-components containing σ R (u) and σ R (v) respectively. Then W u and W v do not contain any folding point and by Lemma 3 If h is a homeomorphism that is pseudo-isotopic to the identity, then R = 0 and the claim of lemma follows. In particular, any homeomorphism h that is pseudo-isotopic to the identity cannot reverse the bar of a sin 1 x -continuum, or reverse a double spiral S ⊂ K s , see Remark 2.4. The next lemma strengthens Lemma 3.1 to the case that W x is allowed to contain folding points.
Lemma 3.5. Let h : K s → K s be a homeomorphism that is pseudo-isotopic to the identity. Let q, p ∈ N 0 be such that h(C q ) C p . Let x be a q-point in the core of K s and let ℓ s l p ∈ C p be such that l = L p (x). Let W x ⊂ ℓ s l p be an arc-component of ℓ s l p containing x. Then h(x) ∈ W x .
Proof. If W x does not contain any folding point the proof follows by Lemma 3.1 for R = 0.
Let W x contain at least one folding point. If x is a folding point, then h(x) = x ∈ W x by The last possibility is that x ∈ (y, z) ⊂ W x , where z ∈ W x is a folding point, y / ∈ W x , i.e., y is a boundary point of W x , and (y, z) does not contain any folding point. Since C 0 is dense in K s , there exists a sequence (W i ) i∈N of arc-components of ℓ s l p such that W i ⊂ C 0 and W i → (y, z] in the Hausdorff metric. Note that for the sequence of p-points (m i ) i∈N , where m i is the midpoint of W i , we have m i → z and L p (m i ) → ∞. Also, for every i large enough, every W i contains a q-point x i with L q (x i ) = L q (x), and for the sequence of q-points (x i ) i∈N we have Proof. We know that h is pseudo-isotopic to σ R for some R ∈ Z; by composing h with σ −R we can assume that R = 0. By Corollary 3.2, h preserves the arc-components, and by Lemma 3.3, preserves the orientation of each arc-component as well.
Take a subsequence such that A n k converges in Hausdorff metric, say to B. Since x, h(x) ∈ B, we have B ⊃ A. Assume by contradiction that B = A. Fix q, p arbitrary such that h(C q ) refines C p , and such that π p (B) = π p (A) and a fortiori, that there is a link ℓ ∈ C p such that ℓ ∩ A = ∅ and π p (ℓ) contains a boundary point of π p (B).
Let d n = max{L p (y) : y is p-point in A n }. If D := sup d n < ∞, then we can pass to the chain C p+D and find that all A n k 's go straight through C p+D , hence the limit is a straight arc as well, stretching from x to h(x), so B = A. Therefore D = ∞, and we can assume without loss of generality that d n k → ∞.
Since the link in ℓ is disjoint from A but π p (ℓ) contains a boundary point of π p (B), the arcs A n k intersects ℓ for all k sufficiently large. Therefore A n k ∩ ℓ separates x n k from h(x n k ); let W n k be a component of A n k ∩ ℓ between x n k and h(x n k ). Since π p (ℓ) contains a boundary point of π p (B), W n k contains at least one p-point for each k. Lemma 3.5 states that there is y n k ∈ W n k such that h(y n k ) ∈ W n k as well, and therefore x n k ≺ y n k , h(y n k ) ≺ h(x n k ) (or y n k ≺ x n k , h(x n k ) ≺ h(y n k )), contradicting that h preserves orientation.
Let us finally prove Theorem 1.3: Proof of Theorem 1.3. Fix R such that h is pseudo-isotopic to σ R . Then σ −R • h is pseudoisotopic to the identity. So renaming σ −R • h to h again, we need to show that h is isotopic to the identity. Clearly t → H(·, t) is a family of maps connecting h to the identity in a single path as t ∈ [0, 1].
We need to show that H is continuous both in x and t, and that H(·, t) is a bijection for all t ∈ [0, 1].
Let z ∈ K s and (z n , t n ) → (z, t). If h(z) = z, then H(z, t) ≡ z, and Proposition 3.6 implies that H(z n , t n ) → z = H(z, t). So let us assume that h(z) = z. The arc A = [z, h(z)] contains no folding point, so by Lemma 2.2, for all x ∈ A, there is ε(x) > 0 and W (x) ∈ N such that B ε(x) (x) contains no p-point of p-level ≥ W (x). By compactness of A, ε := inf x∈A ε(x) > 0 and sup x∈A W (x) < ∞, whence there is m > p + W such that V := π −1 m • π m (A) is contained in an ε-neighborhood of A that contains no p-point.
By Proposition 3.6, there is N such that A n ⊂ V for all n ≥ N, and in fact π m (A n ) → π m (A). It follows that H(z n , t n ) → H(z, t).
To see that x → H(·, t) is injective for all t ∈ [0, 1], assume by contradiction that there is t 0 ∈ [0, 1] and x = y such that H(x, t 0 ) = H(y, t 0 ). Then x and y belong to the same arc-component A, which is the same as the arc-component containing h(x) and h(y). The smallest arc J containing all four point contains no folding point by Corollary 3.4. Therefore there is m such that π m : J → π m (J) is injective, and we can choose an orientation on A such that x < y on J, and π m (x) < π m (y). Since t → π m • H(x, t) is monotone with constant speed depending only on x, we find π m (x) < π m (y) < π m • H(x, t 0 ) = π m • H(y, t 0 ) < π m • h(y) < π m • h(x) This contradicts that h preserves orientation on arc-components, see Lemma 3.3.
Otherwise, say if h(x) > x, there is y < x in the same arc-component as x such that h(y) = x. The map t → H(·, t) moves the arc [y, x] continuously and monotonically to [h(y), h(x)] =