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Dynamics of a family of continued fraction maps

ABSTRACT We explore the dynamics of a 1-parameter family of continued fraction maps of the unit interval. The family contains as special instances the Gauss continued fraction map and the Fibonacci map. We determine the transfer operators of these dynamical maps and prove that the Denjoy–Minkowski measure is a common invariant measure of the family. We show that their analytic invariant measures obey a common functional equation generalizing Lewis’ functional equation and we find a.c. invariant measures for some members of the family. We also discuss a certain involution of this family which sends the Gauss map to the Fibonacci map relating Riemann's zeta function to the so-called Fibonacci zeta function.