The eigenvalues of non-singular transformations (Q759852)

The set of eigenvalues e(T) of a non-singular conservative ergodic transformation T of a separable measure space are considered. They are known to form a Borel subgroup of the circle, with measure zero. The author shows that this is the only metric limitation on the size of e(T). Specifically he shows that for every gauge function \(\rho:[0,1]\to [0,\infty]\) satisfying \(\rho (t)\searrow 0,\) \(\rho (t)/t\nearrow \infty\) as \(t\searrow 0,\) there is a conservative ergodic transformation T of a separable measure space, with a \(\sigma\)-finite invariant measure and such that the \(\rho\)-Hausdorff measure of e(T) is positive. For a sequence \(u_ n\searrow 0\) as \(n\to \infty\) the author defines the terms \(u_ n\)-conservative and \(u_ n\)-dissipative [see \textit{U. Krengel}, Proc. 5th Berkeley Sympos. Math. Stat. Probab. Univ. Calif. 1965/1966 2, Part 2, 415-429 (1967; Zbl 0236.60051)]. The author proves: Theorem 1. If the Hausdorff dimension of e(T) is larger than \(\alpha\in (0,1)\) then T is \(1/n^{1-\alpha}\) dissipative. The second example given shows that for every \(\alpha\in (0,1)\) there is an ergodic \(1/n^{1-\alpha}\) conservative transformation of a separable measure space with a \(\sigma\)-finite invariant measure whose eigenvalues have Hausdorff dimension \(\alpha\). Related results and a lemma on Hausdorff dimension are also given.