A construction of a non-measure-preserving endomorphism using quotient relations and automorphic factors (Q1353623)

The paper gives a new example of an ergodic nonsingular two-to-one endomorphism \(T\) which preserves no equivalent \(\sigma\)-finite measure. \(T\) is the product of the two-sided shift of \textit{T. Hamachi} [Ergodic Theory Dyn. Syst. 1, 273-283 (1981; Zbl 0597.28022)] and a one-sided shift. If \(\gamma\) denotes the Hamachi-measure on the first factor \(Y=\{0,1\}^{\mathbb{R}}\), the measure on \(X=Y\times Y^+\) is of the form \(\mu(C)=\int\rho_y(C\cap\alpha^{-1}y)d\gamma(y)\), where \(\alpha\) is the canonical projection of \(X\) to \(Y\). An algorithm defines \(\rho_y\). The point is that this can be done in such a way that \(T\) is neither exact with respect to \(\mu\) nor is \(\mu\) the product measure of an exact with an automorphic measure.