Inverse Spectra with Two and Three Maps

It is shown that, for any 1 ≤ n < ∞, there exist four maps of the n-dimensional cube to itself such that the limit of any inverse sequence of n-cubes is the limit of some sequence with only these four bonding maps. A universal continuum in the class of all limits of sequences of n-cubes is constructed as the limit of an inverse sequence of n-cubes with one bonding map. All compact sets of trivial shape are represented by using only three maps of the Hilbert cube to itself. Two maps of the closed interval to itself such that any Knaster continuum can be obtained as the limit of an inverse sequence with only these two bonding maps are constructed.