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# Invariant Measures and Natural Extensions

Published:2002-03-01
Printed: Mar 2002
• Andrew Haas
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## Abstract

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
 MSC Classifications: 11J70 - Continued fractions and generalizations [See also 11A55, 11K50] 58F11 - unknown classification 58F1158F03 - unknown classification 58F03

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