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We investigate the question of when an ergodic automorphism T is conjugate to its composition square T 2 , i.e., when does there exist an automor- phism S with the property that ST = T 2 S. This is a non-generic property of automorphisms which seems to be quite exceptional. The situation for ergodic au- tomorphisms having discrete spectrum and automorphisms having the weak closure property is investigated.

Spectral properties of ergodic dynamical systems conjugate to their composition squares