An ergodic automorphism $\bf T$ with singular spectrum of $\bf T^{\otimes n}$ and Lebesgue one of $\bf T^{\otimes (n+1)}$

TL;DR

This paper constructs ergodic automorphisms with tensor powers exhibiting a transition from singular to Lebesgue spectrum as the tensor power increases, depending on a parameter, revealing spectral behavior in ergodic theory.

Contribution

It provides explicit constructions of ergodic automorphisms with controlled spectral types across tensor powers, illustrating a spectrum transition phenomenon.

Findings

Tensor powers of the constructed automorphism have singular spectrum for lower powers.

Higher tensor powers exhibit Lebesgue spectrum, indicating a spectral transition.

The spectrum type depends on the tensor power relative to a parameter $eta$.

Abstract

For any natural $n$, and real $\alpha\geq 0$ we construct an ergodic automorphism $T$ such that its tensor powers $T^{\otimes n}$ have singular spectra if $n\leq 1+\alpha /2$, and Lebesgue if $n\, > 1+\alpha/2$.