It is known (due to Irwin-Solecki) that the so called pseudoarc can be represented as a quotient of a zero-dimensional compact "prespace" under an appropriate equivalence relation (which is an inverse limit of linear graphs), and the automorphisms of this prespace densely embeds into the homeomorphism group of the pseudoarc. Although this embedding is only continuous, not a homeomorphic embedding, we can actually characterize the topology inherited from the homeomorphism group intrinsically, only in terms of the prespace. Using this characterization we show that not all homeomorphisms are conjugate to an automorphism, and we give a second proof to Kwiatkowska's conjecture, namely that there exists a homeomorphism with a dense conjugacy class.

This is joint work with S. Solecki.