Divergent sequences satisfying the linear fractional transformations (Q1210455)

The linear fractional transformation \(T(x)={ax+b\over cx+d}\) with \(a,b,c,d\) real, \(ad-bc=1\) and \(c\neq 0\) maps the extended real axis one- one onto itself. If the fixed points of \(T\) are real, every real sequence satisfying \(x_{n+1}=T(x_ n)\) will converge to one of the fixed points. If \(\lambda\) is this fixed point and \(f\in C(-\infty,\infty)\), then \({1\over N}\sum^ N_ 1f(x_ n)\to f(\lambda)\) as \(N\to\infty\). In this paper the corresponding limit is examined when the fixed points of \(T\) are nonreal (and therefore complex conjugates). If the fixed points are \(\alpha\) and \(\overline\alpha\) then the transformation \(y=T(x)\) may be rewritten \({y-\alpha\over y-\overline\alpha}=e^{i\beta x}{x-\alpha\over x-\overline\alpha}\) for some real \(\beta\) in \((0,2\pi)\). It is shown that if \(\beta\) is not a rational multiple of \(\pi\), then for almost all \(x_ 1\) (in the sense of Lebesgue measure), \({1\over N}\sum^ N_ 1f(x_ n)\) converges to \(\int^ \infty_{- \infty}{Kf(x)dx\over cx^ 2+(d-a)x-b}\), where \(K\) is a certain normalizing constant.