2-manifolds and inverse limits of set-valued functions on intervals

Abstract

Suppose for each n ∈ N, fn : [0, 1] → 2 [0,1] is a function whose graph Γ(fn) = (x, y) ∈ [0, 1] 2 : y ∈ fn(x) is closed in [0, 1] 2 (here 2 [0,1] is the space of non-empty closed subsets of [0, 1]). We show that the generalized inverse limit lim ← − (fn) = (xn) ∈ [0, 1] N : ∀n ∈ N, xn ∈ fn(x n+1 ) of such a sequence of functions cannot be an arbitrary continuum, answering a longstanding open problem in the study of generalized inverse limits. In particular we show that if such an inverse limit is a 2-manifold then it is a torus and hence it is impossible to obtain a sphere.2010 Mathematics Subject Classification. Primary: 54C08, 54E45; Secondary: 54F15, 54F65.

Abstract topological dynamics involving set-valued functions

“…Ingram posed a number of questions in [14] which has motivated a growing number of researchers to work in the area. A list of articles can be found in [13]; more recent examples include [11,17,18,19]. Standard inverse limits in a dynamical setting have been extensively used in areas such as dynamical systems and continua theory [2,8,9,10].…”