"In this paper we study the topological properties of continua which arise as inverse limits on [0; 1] with bonding maps chosen from the permutation family of Markov maps. For such inverse limits, we examine the occurrence of indecomposability, the number of end points in the continuum, and the types of subcontinua present in the continuum. We provide a process for determining the topological structure of the inverse limit generated by a single permutation map, or by the composition of several such maps. Additionally, we show that all such inverse limits are Kelley continua. We will apply these results to study inverse limits on [0,1] with a single bonding map chosen from the one parameter family of logistic mappings. It is known that there is an open and dense subset of the parameter space for which the associated logistic maps have attracting periodic orbits. We show that any continuum generated by such a logistic map is homeomorphic to the inverse limit on [0,1] with some permutation bonding map. We close by providing a sufficient condition for the inverse limit on an interval with a single bonding map to fail to be a Kelley continuum, and applying this information to the logistic family"--Abstract, page iii.
Charatonik, W. J.
Roe, Robert P.
Ingram, W. T. (William Thomas), 1937-
Morgan, Ilene H.
Mathematics and Statistics
Ph. D. in Mathematics
Missouri University of Science and Technology
vii, 74 pages
© 2008 Robbie Allen Beane, All rights reserved.
Dissertation - Open Access
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Beane, Robbie A., "Inverse limits of permutation maps" (2008). Doctoral Dissertations. 1885.