scholarly journals Strong convergence of additive Multidimensional Continued Fraction algorithms

2006 ◽  
Vol 121 (1) ◽  
pp. 1-19
Author(s):  
Kentaro Nakaishi
Keyword(s):  
Strong Convergence ◽  
Continued Fraction ◽  
Multidimensional Continued Fraction

EXPONENTIALLY STRONG CONVERGENCE OF NON-CLASSICAL MULTIDIMENSIONAL CONTINUED FRACTION ALGORITHMS

2002 ◽  
Vol 02 (04) ◽  
pp. 563-586
Author(s):  
KENTARO NAKAISHI
Keyword(s):  
Strong Convergence ◽  
Continued Fraction ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Deterministic Case ◽  
Convergence Properties ◽  
Almost Everywhere ◽  
Multidimensional Continued Fraction

Convergence properties of multidimensional continued fraction algorithms introduced by V. Baladi and A. Nogueira are studied. The paper contains an arithmetic proof of almost everywhere exponentially strong convergence of some two-dimensional Markovian random algorithms and dynamically defined ones. A special three-dimensional deterministic case is also discussed.

scholarly journals A combinatorial approach to products of Pisot substitutions

2015 ◽  
Vol 36 (6) ◽  
pp. 1757-1794 ◽  
Author(s):  
VALÉRIE BERTHÉ ◽  
JÉRÉMIE BOURDON ◽  
TIMO JOLIVET ◽  
ANNE SIEGEL
Keyword(s):  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Continued Fraction ◽  
Combinatorial Approach ◽  
Pisot Numbers ◽  
Algorithmic Framework ◽  
Multidimensional Continued Fraction ◽  
Finite Products ◽  
Dynamical Properties ◽  
Coincidence Conditions

We define a generic algorithmic framework to prove a pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and Jacobi–Perron. Our tools consist in a reformulation of some combinatorial criteria (coincidence conditions), in terms of properties of discrete plane generation using multidimensional (dual) substitutions. We also deduce some topological and dynamical properties of the Rauzy fractals, of the underlying symbolic dynamical systems, as well as some number-theoretical properties of the associated Pisot numbers.

scholarly journals A generalized family of multidimensional continued fractions: TRIP Maps

2014 ◽  
Vol 10 (08) ◽  
pp. 2151-2186 ◽  
Author(s):  
Krishna Dasaratha ◽  
Laure Flapan ◽  
Thomas Garrity ◽  
Chansoo Lee ◽  
Cornelia Mihaila ◽  
...  
Keyword(s):  
Number Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Real Numbers ◽  
Multidimensional Continued Fractions ◽  
The Family ◽  
Before And After ◽  
Multidimensional Continued Fraction

Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

scholarly journals On the periodic writing of cubic irrationals and a generalization of Rédei functions

2015 ◽  
Vol 11 (03) ◽  
pp. 779-799 ◽  
Author(s):  
Nadir Murru
Keyword(s):  
Continued Fraction ◽  
Integer Sequences ◽  
Recurrent Sequences ◽  
Linear Recurrent Sequences ◽  
Algebraic Properties ◽  
Cubic Polynomial ◽  
Multidimensional Continued Fraction

In this paper, we provide a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In particular, for a root α of a cubic polynomial with rational coefficients, we study the Cerruti polynomials [Formula: see text], and [Formula: see text], which are defined via [Formula: see text] Using these polynomials, we show how any cubic irrational can be written periodically as a ternary continued fraction. A periodic multidimensional continued fraction (with pre-period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences.

A NEW SYMMETRIC TWO-DIMENSIONAL ALGORITHM

2006 ◽  
Vol 02 (04) ◽  
pp. 489-498
Author(s):  
PEDRO FORTUNY AYUSO ◽  
FRITZ SCHWEIGER
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Dual Graph ◽  
Plane Curve ◽  
Singularity Theory ◽  
Main Topic ◽  
Two Dimensional ◽  
Dimension 2 ◽  
Multidimensional Continued Fraction ◽  
Fraction Algorithm

Continued fractions are deeply related to Singularity Theory, as the computation of the Puiseux exponents of a plane curve from its dual graph clearly shows. Another closely related topic is Euclid's Algorithm for computing the gcd of two integers (see [2] for a detailed overview). In the first section, we describe a subtractive algorithm for computing the gcd of n integers, related to singularities of curves in affine n-space. This gives rise to a multidimensional continued fraction algorithm whose version in dimension 2 is the main topic of the paper.

scholarly journals Units in families of totally complex algebraic number fields

2004 ◽  
Vol 2004 (45) ◽  
pp. 2383-2400
Author(s):  
L. Ya. Vulakh
Keyword(s):  
Algebraic Number ◽  
Continued Fraction ◽  
Quadratic Field ◽  
Number Fields ◽  
Imaginary Quadratic Field ◽  
Algebraic Number Fields ◽  
Ring Of Integers ◽  
Multidimensional Continued Fraction ◽  
Fundamental Units

Multidimensional continued fraction algorithms associated withGLn(ℤk), whereℤkis the ring of integers of an imaginary quadratic fieldK, are introduced and applied to find systems of fundamental units in families of totally complex algebraic number fields of degrees four, six, and eight.

On almost everywhere strong convergence of multi-dimensional continued fraction algorithms

2000 ◽  
Vol 20 (6) ◽  
pp. 1711-1733 ◽  
Author(s):  
D. M. HARDCASTLE ◽  
K. KHANIN
Keyword(s):  
Strong Convergence ◽  
Continued Fraction ◽  
Arbitrary Dimension ◽  
Computer Assisted ◽  
Numerical Work ◽  
Almost Everywhere

We describe a strategy which allows one to produce computer assisted proofs of almost everywhere strong convergence of Jacobi–Perron type algorithms in arbitrary dimension. Numerical work is carried out in dimension three to illustrate our method. To the best of our knowledge this is the first result on almost everywhere strong convergence in dimension greater than two.

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