Spectral properties of ergodic dynamical systems conjugate to their composition squares
Let $S$ and $T$ be automorphisms of a standard Borel probability space. Some ergodic and spectral consequences of the equation $ST=T^2S$ are given for $T$ ergodic and also when $T^n=I$ for some $n>2$. These ideas are used to construct examples of ergodic automorphisms $S$ with oscillating maximal spectral multiplicity function. Other examples illustrating the theory are given, including Gaussian automorphisms having simple spectra and conjugate to their squares.