Torus-like continua which are not self-covering spaces

For each non-quadratic p-adic integer, p > 2 , we give an example of a torus-like continuum Y (i.e., inverse limit of an inverse sequence, where each term is the 2-torus T 2 and each bonding map is a surjective homomorphism), which admits three 4-sheeted covering maps f 0 : X 0 → Y , f 1 : X 1 → Y , f 2 : X 2 → Y such that the total spaces X 0 = Y , X 1 and X 2 are pair-wise non-homeomorphic. Furthermore, Y admits a 2p-sheeted covering map f 3 : X 3 → Y such that X 3 and Y are non-homeomorphic. In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps.