Homeomorphisms of the pseudo-arc are essentially small (Q582617)

The pseudo-arc is a nondegenerate, hereditarily indecomposable, chainable continuum. Among its many interesting properties are its homogeneity, hereditary equivalence, and the fact that though chainable it admits many effective compact group actions. It is known that every self map of a pseudo-arc is a near-homeomorphism and that every homeomorphism of the pseudo-arc is a product of \(\epsilon\)-homeomorphisms. Recently there has been increasing interest in the dynamics of the pseudo-arc, with Kennedy proving the existence of transitive homeomorphisms. Barge has asked whether there exist such homeomorphisms which are arbitrarily close to the identity. We provide a positive answer to this question and show that any dynamics occurring in homeomorphisms of the pseudo-arc occurs in homeomorphisms which are arbitrarily close to the identity. Specifically, we show that every homeomorphism of the pseudo-arc is conjugate to an \(\epsilon\)-homeomorphism, for each \(\epsilon >0\).