On isotopy and unimodal inverse limit spaces

Abstract

We prove that every self-homeomorphism h : Ks → Ks on the inverse limit space Ks of tent map Ts with slope s ∈ ( √ 2, 2] is isotopic to a power of the shift-homeomorphism σ R : Ks → Ks.

Beyond $0$ and $\infty$: A solution to the Barge Entropy Conjecture

“…It was proven in [5] by Barge, Bruin and Štimac. As a somewhat surprising by-product of its solution (but in agreement with the results described above), it was shown in [8] that the mapping class group of X is Z. This in turn was used to characterize possible values of the topological entropy of homeomorphisms on these spaces [9].…”

Entropy of homeomorphisms on unimodal inverse limit spaces

“…However, among the p-points there are special ones, which we call salient, which are center points of symmetries in C. Homeomorphisms preserve these symmetries to such an extent that it is possible to prove that salient points map close to salient points. In [10] it was shown that x ∈ K s is a folding point if and only if for some p ∈ N there is a sequence of p-points (x k ) k∈N such that x k → x and L p (x k ) → ∞.…”

“…Let us denote by ℓ x p a link of C p which contains the point x. From [19] and [2] (for the finite and infinite critical orbit case, respectively) we have the following proposition: From [6,7] and [10] we can derive of the chain C p . Using this notation we can write Proposition 3.1 in the following way: h(x) ≈ p σ R (x) for every x ∈ E q .…”

“…In this paper, we add to the description of the group of homeomorphisms h : K s → K s for arbitrary s. In [6,7] and [10] it is proven that for s ∈ ( √ 2, 2] and a homeomorphism h : K s → K s , there exists R ∈ Z such that h is isotopic to σ R . Here we study the topological entropy h top (h) and show:…”

“…The one-dimensional situation of Theorem 1.1 is more rigid, and one is inclined to compare it with skew-products where all fiber maps are monotone interval maps. Indeed [10], the rigidity of isotopy implies that h can differ from σ R only at non-folding points, (see Definition 2.2 below). Using Bowen's entropy formula and the notion of sequence entropy, Kolyada, Misiurewicz and Snoha [14,15] proved that for such skew-products the entropy is the same as the entropy of the base map.…”