The Ingram conjecture

Abstract

We prove the Ingram conjecture, ie we show that the inverse limit spaces of tent maps with different slopes in the interval OE1; 2 are nonhomeomorphic. Based on the structure obtained from the proof, we also show that every self-homeomorphism of the inverse limit space of a tent map is pseudo-isotopic, on the core, to some power of the shift homeomorphism. 54H20; 37B45, 37E05

On isotopy and unimodal inverse limit spaces

“…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 self-homeomorphism h : K s → K s is pseudoisotopic 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.…”

“…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. .…”

“…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. This conjecture was posed by Tom Ingram in 1992 for tent maps T s : [0, 1] → [0, 1] with slope ±s, s ∈ [1,2], defined as T s (x) = min{sx, s(1 − x)}. The turning point is c = 1 2 and we denote its iterates by c n = T n s (c).…”

“…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.…”

“…Here we prove the general result. 2], and h : K s → K s is a homeomorphism, then there exists R ∈ Z such that h is isotopic to σ R .…”