Odd-odd continued fraction algorithm

By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among others, it is proved that all the best approximating rationals of odd denominators and odd numerators of an irrational number are given by the principal convergents of the odd-odd continued fraction algorithm and vice versa.