Ergodic properties of triangle partitions

Messaoudi, Ali; Nogueira, Arnaldo; Schweiger, Fritz

doi:10.1007/s00605-008-0065-z

Cited by 22 publications

(35 citation statements)

References 7 publications

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“…This algorithm is reminiscent of the Garrity triangle sequence (Assaf et al [1]; Messaoudi, Nogueira and Schweiger [3]). We find three branches on the triangle with vertices .1; 0/, .0; 1/ and .…”

Section: Another Ideamentioning

confidence: 99%

Variations of the Poincaré Map

Schweiger

2012

Integers

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Section: Another Ideamentioning

confidence: 99%

Variations of the Poincaré Map

Schweiger

2012

Integers

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“…If not, what classes of cubic irrationals will have a periodic triangle sequence under various compositions of maps? As shown in [11] , the original triangle map is ergodic. It is well known that the Mönkemeyer map is ergodic [14].…”

Section: Proofmentioning

confidence: 99%

“…(This is in part the goal of the work of Lagarias in [10].) In this paper, we develop a language by generalizing the triangle map, which was introduced in [6] and further developed in [1], [15] and [11]. This generalization produces a family of 216 multidimensional continued fraction algorithms.…”

Section: Introductionmentioning

confidence: 99%

A generalized family of multidimensional continued fractions: TRIP Maps

Dasaratha

Flapan

Garrity

et al. 2014

Int. J. Number Theory

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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.

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“…We begin by describing the original triangle map, which is a multidimensional continued fraction algorithm, as defined in [10] and further developed in [1,17] . (The triangle map was shown to be 3 erogodic in [14].) We then introduce permutations into the definition of the triangle map, thereby generating a family of 216 multidimensional continued fractions.…”

Section: The Trip Algorithmsmentioning

confidence: 99%

Cubic irrationals and periodicity via a family of multi-dimensional continued fraction algorithms

et al. 2014

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We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers α so that either (α, α 2 ) or (α, α − α 2 ) is purely periodic with respect to an element in the family. These cubic irrationals seem to be quite natural, as we show that, for every cubic number field, there exists a pair (u, u ′ ) with u a unit in the cubic number field (or possibly the quadratic extension of the cubic number field by the square root of the discriminant) such that (u, u ′ ) has a periodic multidimensional continued fraction expansion under one of the maps in the family generated by the initial five maps. These results are built on a careful technical analysis of certain units in cubic number fields and our family of multi-dimensional continued fractions. We then recast the linking of cubic irrationals with periodicity to the linking of cubic irrationals with the construction of a matrix with nonnegative integer entries for which at least one row is eventually periodic.

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Ergodic properties of triangle partitions

Cited by 22 publications

References 7 publications

Variations of the Poincaré Map

Variations of the Poincaré Map

A generalized family of multidimensional continued fractions: TRIP Maps

Cubic irrationals and periodicity via a family of multi-dimensional continued fraction algorithms

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