Fixed point set characterizations of Peano continua and absolute retracts

We extend the notion of absolute fixed point sets to the setting of continuum-valued maps whose point images have small diameters. We demonstrate that the resulting class of spaces (ε-MAFS) coincides with the class of absolute fixed point sets and with the class of absolute retracts in the one-dimensional and planar settings, but that the class of n-dimensional absolute fixed point sets is strictly contained in ε-MAFS for all n>1. Moreover, we will give two characterizations for the class of Peano continua in terms of ε-MAFS.