The Bi-sequences of Approximation Coefficients for Gauss-like and Renyi-like Maps on the Interval

Abstract

We will establish several arithmetic and geometric properties regarding the bi-sequences of approximation coefficients (BAC) associated with the two one-parameter families of piecewise-continuous Möbius transformations introduced by Haas and Molnar. The Gauss and Renyi maps, which lead to the expansions of irrational numbers on the interval as regular and backwards continued fractions, are realized as special cases. The results are natural generalizations of theorems from Diophantine approximation.