About the existence of the thermodynamic limit for some deterministic sequences of the unit circle (Q5926146)

A problem that arose in an earlier work of the author, \textit{G. Turchetti} and \textit{S. Vaienti} [J. Stat. Phys. 75, No. 1-2, 167-187 (1994; Zbl 0833.58016)] concerns the almost-everywhere behaviour of the sequence \(x_n=h(\lambda^n-\lambda^{-n})\) mod \(1\), viewed as an element of \([-1/2,1/2)\). The main theorem here states that \((1/n)\ln|x_n|\to 0\) for Lebesgue almost-every \((h,\lambda)\in{\mathbb R}_{+}\times(1,\infty)\).