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

Abstract

We show that in the set Ω=ℝ+×(1,+∞)⊂ℝ+2, endowed with the usual Lebesgue measure, for almost all (h,λ)∈Ω the limit limn→+∞(1/n)ln|h(λn−λ−n)mod[-12,12)| exists and is equal to zero. The result is related to a characterization of relaxation to equilibrium in mixing automorphisms of the two-torus. It is nothing but a curiosity, but maybe you will find it nice.