Towards the complete classification of generalized tent maps inverse limits
Given numbers \(a,b\in [0,1]\), the generalized tent map \(f_{(a,b)}:[0,1]\rightarrow [0,1]\) is the set-valued function with the graph being the union of the segment (possibly degenerate) from \((0,0)\) to \((a,b)\) and the segment (possibly degenerate) from \((a,b)\) to \((1,0)\). The point \((a,b)\) is called the top point of \(f_{(a,b)}\). The inverse limit of a sequence of intervals \([0,1]\) with unique bonding map \(f_{(a,b)}\) is denoted by \(K_{(a,b)}\). Even when the mappings \(f_{(a,b)}\) are very simple, the space \(K_{(a,b)}\) can be extremly complicated. For example, it is not known what the space \(K_{(0,1)}\) looks like. In this paper the authors study continua of the form \(K_{(a,b)}\). They give a complete description of \(K_{(a,b)}\) when \((a,b)\) belongs to some specific subsets of \([0,1]^{2}\). They define a countable family of curves \(C_{n}\) in \([0,1]^{2}\) and they prove that \(K_{(a,b)}\) and \(K_{(c,d)}\) are homeomorphic if they belong to the same \(C_{n}\). The authors also find some new properties of \(K_{(0,1)}\).
scientific article; zbMATH DE number 6121483