224 research outputs found
Multidimensional continued fractions and a Minkowski function
The Minkowski Question Mark function can be characterized as the unique
homeomorphism of the real unit interval that conjugates the Farey map with the
tent map. We construct an n-dimensional analogue of the Minkowski function as
the only homeomorphism of an n-simplex that conjugates the piecewise-fractional
map associated to the Monkemeyer continued fraction algorithm with an
appropriate tent map.Comment: 17 pages, 3 figures. Revised version according to the referee's
suggestions. Proof of Lemma 2.3 more detailed, other minor modifications. To
appear in Monatshefte fur Mathemati
Multidimensional continued fractions, dynamical renormalization and KAM theory
The disadvantage of `traditional' multidimensional continued fraction
algorithms is that it is not known whether they provide simultaneous rational
approximations for generic vectors. Following ideas of Dani, Lagarias and
Kleinbock-Margulis we describe a simple algorithm based on the dynamics of
flows on the homogeneous space SL(2,Z)\SL(2,R) (the space of lattices of
covolume one) that indeed yields best possible approximations to any irrational
vector. The algorithm is ideally suited for a number of dynamical applications
that involve small divisor problems. We explicitely construct renormalization
schemes for (a) the linearization of vector fields on tori of arbitrary
dimension and (b) the construction of invariant tori for Hamiltonian systems.Comment: 51 page
Linear recurrence sequences and periodicity of multidimensional continued fractions
Multidimensional continued fractions generalize classical continued fractions
with the aim of providing periodic representations of algebraic irrationalities
by means of integer sequences. However, there does not exist any algorithm that
provides a periodic multidimensional continued fraction when algebraic
irrationalities are given as inputs. In this paper, we provide a
characterization for periodicity of Jacobi--Perron algorithm by means of linear
recurrence sequences. In particular, we prove that partial quotients of a
multidimensional continued fraction are periodic if and only if numerators and
denominators of convergents are linear recurrence sequences, generalizing
similar results that hold for classical continued fractions
On p-adic Multidimensional Continued Fractions
Multidimensional continued fractions (MCFs) were introduced by Jacobi and
Perron in order to generalize the classical continued fractions. In this paper,
we propose an introductive fundamental study about MCFs in the field of the
--adic numbers . First, we introduce them from a formal point
of view, i.e., without considering a specific algorithm that produces the
partial quotients of a MCF, and we perform a general study about their
convergence in . In particular, we derive some conditions about
their convergence and we prove that convergent MCFs always strongly converge in
contrarily to the real case where strong convergence is not ever
guaranteed. Then, we focus on a specific algorithm that, starting from a
--tuple of numbers in , produces the partial quotients of the
corresponding MCF. We see that this algorithm is derived from a generalized
--adic Euclidean algorithm and we prove that it always terminates in a
finite number of steps when it processes rational numbers
Combinatorial properties of multidimensional continued fractions
The study of combinatorial properties of mathematical objects is a very
important research field and continued fractions have been deeply studied in
this sense. However, multidimensional continued fractions, which are a
generalization arising from an algorithm due to Jacobi, have been poorly
investigated in this sense, up to now. In this paper, we propose a
combinatorial interpretation of the convergents of multidimensional continued
fractions in terms of counting some particular tilings, generalizing some
results that hold for classical continued fractions
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