The lap-counting function for linear mod one transformations I: explicit formulas and renormalizability

Abstract

AbstractLinear mod one transformations are the maps of the unit interval given by fβα(x) = βx + α (mod 1), with β > 1 and 0 ≤ α < 1. The lap-counting function is the function where the lap number Ln essentially counts the number of monotonic pieces of the nth iterate . We derive an explicit factorization formula for Lβα(z) which directly shows that Lβα(z) is a function meromorphic in the open unit disk {z: |z| < 1} and analytic in the open disk {z: |z| < 1/β}, with a simple pole at z = 1/β.Comparison with a known formula for the Artin—Mazur—Ruelle zeta function ζβ,α(z) of fβα shows that Lβα(z) and ζβ,α(z) have identical sets of singularities in the disk {z: |z| < 1}. We derive two more factorization formulae for Lβ,α(z) valid for certain parameter ranges of (β, α). When 1 < α + β ≤ 2, there is sometimes a ‘renormalization’ structure of such maps present, which has previously been studied in connection with simplified models for the Lorenz attractor. In the case that fβα is non-trivially renormalizable, we obtain a factorization formula for Lβα(z). For (β, α) in a region contained in 2 < α + β ≤ 3 we obtain a factorization formula which relates Lβα(z) to a ‘rescaled’ lap-counting function from the region 1 < α + β ≤ 2. The various factorizations exhibit certain singularities of Lβα(z) on the circle |z| = 1/β. These singularities are related to topological dynamical properties of fβ,α. In parts II and III we show that these comprise the complete set of such singularities on the circle |z| = 1/β.