Continued fractions with sl(2, z)-branches: Combinatorics and entropy

We study the dynamics of a family Kα of discontinuous interval maps whose (infinitely many) branches are Möbius transformations in SL(2, Z) and which arise as the critical-line case of the family of (a, b)-continued fractions. We provide an explicit construction of the bifurcation locus EKU for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of Kα is analytic outside the bifurcation set but not differentiable at points of EKU and that the entropy is monotone as a function of the parameter. Finally, we prove that the bifurcation set is combinatorially isomorphic to the main cardioid in the Mandelbrot set, providing one more entry to the dictionary developed by the authors between continued fractions and complex dynamics.