Abstract
This chapter introduces the continued fraction decomposition for real numbers and develops the basic properties of the continued fraction. The relationship between the continued fraction expansion and the Gauss map viewed as a measure-preserving transformation is described, and an elementary proof of ergodicity of the Gauss map is given. The relationship between continued fractions and Diophantine approximation is introduced, and some of the properties of badly approximable numbers are described. The continued fraction map will be revisited in Chapter 9 via the geodesic flow on a homogeneous space.
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© 2011 Springer-Verlag London Limited
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Einsiedler, M., Ward, T. (2011). Continued Fractions. In: Ergodic Theory., vol 259. Springer, London. https://doi.org/10.1007/978-0-85729-021-2_3
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DOI: https://doi.org/10.1007/978-0-85729-021-2_3
Publisher Name: Springer, London
Print ISBN: 978-0-85729-020-5
Online ISBN: 978-0-85729-021-2
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