We study the dynamics of a family K_alpha of discontinuous interval mapswhose (infinitely many) branches are Moebius transformations in SL(2, Z), andwhich arise as the critical-line case of the family of (a, b)-continuedfractions. We provide an explicit construction of the bifurcation locus E_KUfor this family, showing it is parametrized by Farey words and it has Hausdorffdimension zero. As a consequence, we prove that the metric entropy of K_alphais analytic outside the bifurcation set but not differentiable at points ofE_KU, and that the entropy is monotone as a function of the parameter. Finally,we prove that the bifurcation set is combinatorially isomorphic to the maincardioid in the Mandelbrot set, providing one more entry to the dictionarydeveloped by the authors between continued fractions and complex dynamics.
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