turkmath.org

I will introduce the involution of the real line induced by the outer automorphism of the extended modular group PGL(2,Z). It comes from the abstract tree automorphism of the planar trivalent tree, which twists every vertex of the tree. It can be described as a basic continued fraction transformation which modifies the infinity of the continued fraction. This involution is discontinuous at rationals but satisfies a surprising collection of functional equations. It preserves the set of real quadratic irrationalities mapping them in a highly non-obvious way to each other. More generally, it preserves set-wise the orbits of the modular group thereby inducing an involution of the moduli space of real rank-two lattices. It conjugates (though not topologically) the Gauss' continued fraction map to an intermittent dynamical system on the unit interval with an infinite invariant measure. The transfer operator (resp. the functional equation) naturally associated to this dynamical system is closely related to the Mayer transfer operator (resp. the Lewis' functional equation). We prove that its derivative exists and vanishes almost everywhere. It is conjectured that algebraic numbers of degree at least three are mapped to transcendental numbers under this involution.