Generic inverse limits with set-valued functions

被引:0
|
作者
Greenwood, Sina [1 ]
Kennedy, Judy [2 ]
机构
[1] Univ Auckland, Dept Math, Private Bag 92019, Auckland, New Zealand
[2] Lamar Univ, Dept Math, POB 10047, Beaumont, TX 77710 USA
关键词
Generalized inverse limit; Sub-generalized inverse limit; Inverse limit with set-valued functions; Hereditarily disconnected; Generic; Pseudoarc; SEMICONTINUOUS BONDING FUNCTIONS; GRAPHS; MAPS;
D O I
10.1016/j.topol.2022.108142
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Suppose that for each i >= 0, Iiis an interval, and for each i >= 1, Piis a pseudoarc contained in Ii-1 x I-i such that pi P-i-1(i)= Ii-1 and pi P-i(i)= I-i(pi(i-1)and pi(i)denote the respective projections of P-i to the intervals Ii-1 and I-i). Then for each i >= 1, there is a surjective upper semicontinuous map f(i) : Ii -> 2(Ii-1)such that the graph Gamma(f(i)) = P-i. We prove that the inverse limit space (lim) under left arrow (I-i, f(i)): ={(x(0), x(1),...) is an element of Pi (infinity)(i=0) I-i : for each i >= 1, x(i-1) is an element of f(i)(x(i))} is hereditarily disconnected (i.e., no closed nondegenerate sub-generalized inverse limit is connected). It does contain, however, nondegenerate continua. Furthermore, in the space of all such inverse limits in the Hilbert cube, those inverse limits that are formed with pseudoarc bonding maps (we call them pseudoarc generalized inverse limits) form a dense G(delta)-set. It follows that such inverse limits generated by set-valued functions are generic, and that a generic inverse limit generated by set-valued functions is hereditarily disconnected with no isolated points. (c) 2022 Elsevier B.V. All rights reserved.
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页数:15
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