38,974 research outputs found
Commensurable continued fractions
We compare two families of continued fractions algorithms, the symmetrized
Rosen algorithm and the Veech algorithm. Each of these algorithms expands real
numbers in terms of certain algebraic integers. We give explicit models of the
natural extension of the maps associated with these algorithms; prove that
these natural extensions are in fact conjugate to the first return map of the
geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost
every real number has an infinite number of common approximants for both
algorithms.Comment: 41 pages, 10 figure
Cross sections for geodesic flows and \alpha-continued fractions
We adjust Arnoux's coding, in terms of regular continued fractions, of the
geodesic flow on the modular surface to give a cross section on which the
return map is a double cover of the natural extension for the \alpha-continued
fractions, for each in (0,1]. The argument is sufficiently robust to
apply to the Rosen continued fractions and their recently introduced
\alpha-variants.Comment: 20 pages, 2 figure
Formulas for Continued Fractions. An Automated Guess and Prove Approach
We describe a simple method that produces automatically closed forms for the
coefficients of continued fractions expansions of a large number of special
functions. The function is specified by a non-linear differential equation and
initial conditions. This is used to generate the first few coefficients and
from there a conjectured formula. This formula is then proved automatically
thanks to a linear recurrence satisfied by some remainder terms. Extensive
experiments show that this simple approach and its straightforward
generalization to difference and -difference equations capture a large part
of the formulas in the literature on continued fractions.Comment: Maple worksheet attache
Tanaka-Ito -continued fractions and matching
Two closely related families of -continued fractions were
introduced in 1981: by Nakada on the one hand, by Tanaka and Ito on the other
hand. The behavior of the entropy as a function of the parameter has
been studied extensively for Nakada's family, and several of the results have
been obtained exploiting an algebraic feature called matching. In this article
we show that matching occurs also for Tanaka-Ito -continued
fractions, and that the parameter space is almost completely covered by
matching intervals. Indeed, the set of parameters for which the matching
condition does not hold, called bifurcation set, is a zero measure set (even if
it has full Hausdorff dimension). This property is also shared by Nakada's
-continued fractions, and yet there also are some substantial
differences: not only does the bifurcation set for Tanaka-Ito continued
fractions contain infinitely many rational values, it also contains numbers
with unbounded partial quotients
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