| This dissertation studies r--continued fraction expansions, which are a type of generalized continued fraction expansions. We show that every real number has a valid r--continued fraction expansion. This result extends previous results about regular continued fraction expansions and N--continued fractions expansions.;Parry proved that a sequence is admissible as the beta--expansion of a real number if and only if all of the shifts of the sequence are dominated lexicographically by the sequence obtained from the betaexpansion of 1. We prove a similar result for r--continued fraction expansions. We prove that a sequence is admissible, that is, it comes from the r--continued fraction map, if and only if all of its shifts are alternating--lexicographically less than the sequence obtained from the r--continued fraction expansion of 1. We also prove that if the r--continued fraction expansion of a number is finite or periodic, then r must be algebraic.;Finally, we show that the r--continued fraction map admits an absolutely continuous invariant measure that is equivalent to Lebesgue measure. |
