35,128 research outputs found
A geometric representation of continued fractions
Inspired by work of Ford, we describe a geometric representation of real and complex continued fractions by chains of horocycles and horospheres in hyperbolic space. We explore this representation using the isometric action of the group of Moebius transformations on hyperbolic space, and prove a classical theorem on continued fractions
Even-integer continued fractions and the Farey tree
Singerman introduced to the theory of maps on surfaces an object that is a universal cover for any map. This object is a tessellation of the hyperbolic plane together with a certain subset of the ideal boundary. The 1-skeleton of this tessellation comprises the edges of an infinite tree whose vertices belong to the ideal boundary. Here we show how this tree can be used to give a beautiful geometric representation of even-integer continued fractions. We use this representation to prove some of the fundamental theorems on even-integer continued fractions that are already known, and we also prove some new theorems with this technique, which have familiar counterparts in the theory of regular continued fractions
A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs
We study the two-dimensional continued fraction algorithm introduced in
\cite{garr} and the associated \emph{triangle map} , defined on a triangle
. We introduce a slow version of the triangle map, the
map , which is ergodic with respect to the Lebesgue measure and preserves an
infinite Lebesgue-absolutely continuous invariant measure. We discuss the
properties that the two maps and share with the classical Gauss and
Farey maps on the interval, including an analogue of the weak law of large
numbers and of Khinchin's weak law for the digits of the triangle sequence, the
expansion associated to . Finally, we confirm the role of the map as a
two-dimensional version of the Farey map by introducing a complete tree of
rational pairs, constructed using the inverse branches of , in the same way
as the Farey tree is generated by the Farey map, and then, equivalently,
generated by a generalised mediant operation.Comment: 32 pages. The main results have slightly changed due to a mistake in
the previous versio
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