We study ergodic properties of a family of interval maps that are given as the fractional parts of certain real M\"obius transformations. Included are the maps that are exactly $n$-to-$1$, the classical Gauss map and the Renyi or backward continued fraction map. A new approach is presented for deriving explicit realizations of natural automorphic extensions and their invariant measures.

Keywords:

Continued fractions, interval maps, invariant measures Continued fractions, interval maps, invariant measures

MSC Classifications:

11J70, 58F11, 58F03 show english descriptions Continued fractions and generalizations [See also 11A55, 11K50]
unknown classification 58F11
unknown classification 58F03 11J70 - Continued fractions and generalizations [See also 11A55, 11K50]
58F11 - unknown classification 58F11
58F03 - unknown classification 58F03

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