Weakly confluent, 2-to-1 maps on hereditarily indecomposable continua

Abstract

One of the most important open questions today in the study of exactly 2 2 -to- 1 1 maps is whether or not such a map can be defined on the pseudoarc. We show that there is no weakly confluent 2 2 -to- 1 1 map defined on the pseudoarc. More precisely, it is shown that any reduced weakly confluent 2 2 -to- 1 1 map defined on a hereditarily indecomposable metric continuum must be a confluent local homeomorphism. It follows from this that if there is a weakly confluent 2 2 -to- 1 1 map from a hereditarily indecomposable continuum X X onto a continuum Y Y , then neither X X nor Y Y can be treelike.