Monotone Decompositions of Inverse Limit Spaces Based on Finite Graphs
A Θn,L graph is defined to be a compact, connected, locally connected metric space which is not separated into more than n components by any subcontinuum and no subcontinuum is separated into more than L components by any of its subcontinua. If X is a Θn,L graph and f is a continuous surjection of X onto X, then the inverse limit space (X,f) is a Θn continuum (not necessarily locally connected). Furthermore (X,f) admits a unique minimal monotone, upper semicontinuous decomposition such that the quotient space (X,f)/ is a Θn,L graph if and only if (X,f) contains no indecomposable subcontinua with nonempty interior.
R. P. Roe, "Monotone Decompositions of Inverse Limit Spaces Based on Finite Graphs," Topology and its Applications, Elsevier, Jan 1990.
The definitive version is available at http://dx.doi.org/10.1016/0166-8641(90)90040-9
Mathematics and Statistics
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