On the r-Continued Fraction Expansions of Real Numbers

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 β–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 βexpansion 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.

Author

• Mehmetaj, Erblin

Language

• en

Date created

• 2014

Type of Work

• Dissertation

License

• All rights reserved

Rights statement

• In Copyright

GW Unit

• Mathematics

Degree

• Ph.D.

Advisor

• Robinson, Jr., E. A.

Committee Member(s)

• Harizanov, Valentina
• Abrams, Lowell
• Bourla, Avraham
• Ullman, Daniel

Persistent URL

•  https://scholarspace.library.gwu.edu/etd/jw827b77b