On circle-valued cocycles of an ergodic measure preserving transformation
Let \(\tau\) be an invertible ergodic measure preserving transformation of a probability space (X,\(\mu)\), and let f be a measurable function from X into the circle \({\mathbb{T}}\). If there is a measurable g: \(X\to {\mathbb{T}}\) and a scalar \(\lambda\), \(| \lambda | =1\), with \(f(x)=\lambda g(x)/g(\tau (x))\) for \(\mu\) almost all x, then f is called to be a projective coboundary of \(\tau\). The paper under review contains necessary and sufficient conditions for a measurable f: \(X\to {\mathbb{T}}\) to be a projective coboundary of \(\tau\). It is proved, in particular, that if there exists a sequence \(\{a_ n\}\) of complex numbers of modulus \(\leq 1\) such that \[ (1/N)\sum^{N- 1}_{n=0}a_ n\int f^{(n)}(x) d\mu (x) \] does not tend to 0, then f is a projective coboundary of \(\tau\). Here \(f^{(n)}(x)=f(x)f(\tau (x))...f(\tau^{n-1}(x)),\quad f^{(0)}(x)=1.\) Two applications are given.
invertible ergodic measure preserving transformation
scientific article; zbMATH DE number 4063450