Quasi-invariant measures on the torus \(\mathbb{T}^d\) (Q1884400)

Let \(T: X\to X\) be a dynamical system on a compact metric space \(X\). The author considers the question of finding probability measures \(\mu\), so that \(T^{-1}(\mu)\) is absolutely continuous with respect to \(\mu\) and its Radon-Nikodým derivative is a known continuous strictly positive function \(h\). First, it is shown that many solutions of this problem may exist (Theorem 1), including translations of the torus. The main result is a generalization of a result due to \textit{J.-P. Conze} and \textit{Y. Guivarc'h} [Colloq. Math. 84--85, Pt. 2, 457--480 (2000; Zbl 0963.37009)]. If \(T\) is a translation of the \(d\)-dimensional torus, then the set of strictly positive continuous functions \(h\) satisfying \(\int_{\mathbb{T}^d}\ln h(x)\,dx= 0\) contains a dense \(G_\delta\)-set which has a unique solution \(\mu\).