Mapping chainable continua onto dendroids

We prove that every chainable continuum can be mapped into a dendroid such that all point-inverses consist of at most three points. In particular, it follows that there exists a finite-to-one map from a hereditarily indecomposable continuum (the pseudoarc) onto a hereditarily decomposable continuum. This answers a question by J. Krasinkiewicz.