Cohomology of step functions under irrational rotations
Let \(X=[0,1]\) with addition mod 1 and let \(\theta\) be an irrational number in X. The author studies step functions f of X to the unit circle for which there exist constants \(\lambda\), \(| \lambda | =1\), and measurable functions g such that \((1)\quad g(x+\theta)=\lambda f(x)g(x)\) (that is, f is a multiple of a coboundary). Generalizing results of \textit{W. A. Veech} [Trans. Am. Math. Soc. 140, 1-33 (1969; Zbl 0201.056)], \textit{M. Stewart} [Acta Math. Acad. Sci. Hung. 37, 185-221 (1981; Zbl 0475.10040)] and \textit{K. Petersen} [Compos. Math. 26, 313-317 (1973; Zbl 0269.10030)], the author gives explicit formulas for solutions g of (1) for \(f=f_{t,s}=\exp (2\pi is\chi_{[1-t,1)})\) and \(t=k\theta \quad (mod 1).\) For a bigger set of t's it is shown that \(f_{t,s}\) is a multiple of a coboundary. For \(\theta\) with bounded partial quotients in its continued fraction expansion the author describes a family of step functions f (with finite number of points of discontinuity) for which (1) has no solution.
scientific article; zbMATH DE number 3998140