We show that there are (1) nonhomogeneous metric continua that admit minimal noninvertible maps but have the fixed point property for homeomorphisms, and (2) nonhomogeneous metric continua that admit both minimal noninvertible maps and minimal homeomorphisms. The former continua are constructed as quotient spaces of the torus or as subsets of the torus, the latter are constructed as subsets of the torus.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm95-1-10,
author = {Henk Bruin and Sergi\v\i\ Kolyada and L'ubom\'\i r Snoha},
title = {Minimal nonhomogeneous continua},
journal = {Colloquium Mathematicae},
volume = {96},
year = {2003},
pages = {123-132},
zbl = {1082.37022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm95-1-10}
}
Henk Bruin; Sergiǐ Kolyada; L'ubomír Snoha. Minimal nonhomogeneous continua. Colloquium Mathematicae, Tome 96 (2003) pp. 123-132. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm95-1-10/