We will introduce the codenominator function F that extends the Fibonacci sequence to the index set of positive rational numbers, and show that one can express Dyer’s outer automorphism of PGL(2, Z) in terms of F. This automorphism can be viewed as an automorphism group of the trivalent tree. The real equivariant modular function Jimm (J) on the real line is defined via the codenominator. J relates the Stern-Brocot tree to the Bird tree. Jimm induces an involution of the moduli space of rank-2 pseudolattices and is related to the arithmetic of real quadratic irrationals. Finally, we will explain how J is related to the quantizations of real numbers recently found by Morier-Genoud and Ovsienko and introduce a new conjugate pair of quantizations discovered recently by Mustafa Topkara.