Concerning preservation of chainability upon taking a preimage under

If X is an arc in the complex plane C with 0 as an endpoint, then the preimage of X under f ( z ) = z 2 is also an arc, and the endpoints of f − 1 ( X ) are the points in the preimage of the nonzero endpoint of X. In this paper, the author explores necessary and sufficient conditions under which a chainable continuum in C has chainable preimage under f. The paper contains an example of a chainable continuum X (the simple three-fold Knaster continuum) embedded in the complex plane in such a way that 0 is an endpoint of X and the preimage of X under the square map is not chainable.