The Ingram conjecture. (English) Zbl 1285.54030

Continua as inverse limits have been studied for a long time. One reason for such intense research in this area is the fact that inverse sequences with very simple spaces and simple bonding maps can produce as inverse limits extremely complicated continua. This may happen even in the case where all the spaces are unit intervals \([0,1]\) and all the bonding maps are the same. The famous Brouwer-Janiszewski-Knaster continuum, that can be presented as the inverse limit of the inverse sequence of closed unit intervals \([0,1]\) with the standard tent bonding map \(T_{2}:[0,1]\to [0,1]\), \(T_2(t)=\min\{2t,2(1-t)\}\) is an example. One of the most famous problems in the area is the one about classifying the inverse limits of closed unit intervals \([0,1]\) with a single tent map \(T_s\), \(1\leq s\leq 2\), \(T_s(t)=\min\{st,s(1-t)\}\), on \([0,1]\) as the bonding map.
In 1991, W.T. Ingram stated the following conjecture for tent maps:
If \(1\leq s<s'\leq 2\), then the corresponding inverse limit spaces \(\displaystyle \lim_{\leftarrow}([0,1],T_s)\) and \(\displaystyle \lim_{\leftarrow}([0,1],T_{s'})\) are nonhomeomorphic.
This conjecture became known as the Ingram conjecture and it has received a large amount of attention in the last 23 years. It generated a large number of articles, in which certain special cases of the conjecture were proved. Finally, the conjecture was proved in 2009 by the authors of the paper under review. Their proof is presented in this paper.
The following is the main result of the paper.
{ Theorem 1.1.} If \(1\leq s<s'\leq 2\), then the corresponding inverse limit spaces \(\displaystyle \lim_{\leftarrow}([0,1],T_s)\) and \(\displaystyle \lim_{\leftarrow}([0,1],T_{s'})\) are nonhomeomorphic.


54H20 Topological dynamics (MSC2010)
37B45 Continua theory in dynamics
37E05 Dynamical systems involving maps of the interval
54F15 Continua and generalizations
Full Text: DOI arXiv


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