Ergodicity induces a maximal spectral type equivalent to the Lebesgue measure (Q1386730)

Let \(T\) be an ergodic automorphism of a Lebesgue space \((X, {\mathcal A},m)\). The \(\sigma\)-algebra \({\mathcal A}\) will be endowed with the distance \(d(A,B)= m(A \Delta B)\). If \(m(A) >0\), denote by \(T_A\) the induced automorphism on \(A\). It is known that for a dense family of sets \(A\in {\mathcal A}\) the induced automorphism \(T_A\) is mixing [\textit{N. A. Friedman} and \textit{D. S. Ornstein}, Adv. Math. 10, 147-163 (1973; Zbl 0246.28011)]. The author obtains a much stronger result by proving that the maximal spectral type of \(T_A\) is equivalent to Lebesgue measure for a dense family of \(A\)'s. An important ingredient of the proof is a skew product extension of the \((1- \delta, \delta)\) Bernoulli scheme (for a small \(\delta >0)\) by the given \(T\). Next an approximation technique associates with a given spectral measure \(\sigma\) its ``scattered'' counterpart \((\sigma)_\delta\), which is an absolutely continuous measure with density \[ \int {1-\bigl | (1-\delta) e^{-i\tau} +\delta e^{-2i \tau} \bigr |^2 \over \biggr| 1-e^{it}\bigl((1- \delta) e^{-i\tau} +\delta e^{-2i\tau} \bigr)\biggr |^2} d \sigma (\tau). \] The paper is concluded by a number of open questions. In particular, it is not known whether there exists a zero entropy automorphism \(T\) for which the family of \(A\)'s such that the maximal spectral type of \(T_A\) is Lebesgue contains a dense \(G_\delta\) subset of \({\mathcal A}\). The same question for the property of mixing was asked by A. del Junco and D. J. Rudolph.