Three-dimensional symmetric shift radix systems

Huszti, Andrea; Scheicher, Klaus; Surer, Paul; Thuswaldner, Jörg Μ.

doi:10.4064/aa129-2-2

Cited by 8 publications

(7 citation statements)

References 13 publications

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“…In [7, Conjecture 13], Conjecture 1.1 has been formulated for 1-SRS. For 1 ¼ 1/2, it is shown to be true for d # 3 in [1,10]. Indeed, (1/2)-SRS with finiteness property has been completely characterized for d # 3.…”

Section: Introductionmentioning

confidence: 93%

Contractivity of three-dimensional shift radix systems with finiteness property

Brunotte¹,

Kirschenhofer

Thuswaldner³

2012

Journal of Difference Equations and Applications

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Dedicated to Professor Attila Pethő on the occasion of his 60th birthday Let d $ 1 be an integer and r ¼ ðr 1 ; . . . ; r d Þ [ R d . We define the shift radix system t r :The shift radix system t r has the finiteness property if each a [ Z d is eventually mapped to 0 under iterations of t r . The mapping t r can be written as t r ðaÞ ¼ RðrÞa þ vðaÞ, where R(r) is a d £ d matrix and v is a correction term. It has been conjectured that the fact that t r has the finiteness property implies that all eigenvalues of R(r) are strictly smaller than one in modulus.The aim of the present paper is to prove this conjecture for the case d ¼ 3.

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Section: Introductionmentioning

confidence: 93%

Contractivity of three-dimensional shift radix systems with finiteness property

Brunotte¹,

Kirschenhofer

Thuswaldner³

2012

Journal of Difference Equations and Applications

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“…The sets D 0 n and D n have been extensively studied in the last years (cf. [2,3,4,5,7,11,14,19,24]). In Figure 1, an approximation of D 0 2 is shown.…”

Section: Shift Radix Systems and Finite Beta-expansionsmentioning

confidence: 99%

Beta-expansions of -adic numbers

Scheicher

Sirvent

Surer

2014

Ergod. Th. Dynam. Sys.

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“…It turns out that the characterization of the (accordingly defined) finiteness property is easier in this case and complete results have been achieved for dimension two (see [15]) and three (see [39]). As mentioned above, for SRS it is conjectured that SRS tiles always induce weak tilings.…”

Section: Variants Of Shift Radix Systemsmentioning

confidence: 99%

Shift Radix Systems - A Survey

Kirschenhofer,

Thuswaldner

2013

Preprint

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Let d ≥ 1 be an integer and r = (r 0 , . . . , r d−1 ) ∈ R d . The shift radix system τr : Z d → Z d is defined by τr(z) = (z 1 , . . . , z d−1 , − rz ) t (z = (z 0 , . . . , z d−1 ) t ).τr has the finiteness property if each z ∈ Z d is eventually mapped to 0 under iterations of τr. In the present survey we summarize results on these nearly linear mappings. We discuss how these mappings are related to well-known numeration systems, to rotations with round-offs, and to a conjecture on periodic expansions w.r.t. Salem numbers. Moreover, we review the behavior of the orbits of points under iterations of τr with special emphasis on ultimately periodic orbits and on the finiteness property. We also describe a geometric theory related to shift radix systems. Contents 1. Introduction 2. The relation between shift radix systems and numeration systems 2.1. Shift radix systems and beta-expansions 2.2. Shift radix systems and canonical number systems 2.3. Digit expansions based on shift radix systems 3. Shift radix systems with periodic orbits: the sets D d and the Schur-Cohn region 3.1. The sets D d and their relations to the Schur-Cohn region 3.2. The Schur-Cohn region and its boundary 4. The boundary of D 2 and discretized rotations in R 2 4.1. The case of real roots 4.2. Complex roots and discretized rotations 4.3. A parameter associated with the golden ratio 4.4. Quadratic irrationals that give rise to rational rotations 4.5. Rational parameters and p-adic dynamics 4.6. Newer developments 5. The boundary of D d and periodic expansions w.r.t. Salem numbers 5.1. The case of real roots 5.2. The conjecture of Klaus Schmidt on Salem numbers 5.3. The expansion of 1 5.4. The heuristic model of Boyd for shift radix systems 6. Shift radix systems with finiteness property: the sets D (0) d 6.1. Algorithms to determine D (0) d 6.2. The finiteness property on the boundary of E d 7. The geometry of shift radix systems 7.1. SRS tiles 7.2. Tiling properties of SRS tiles 7.3. SRS tiles and their relations to beta-tiles and self-affine tiles 8. Variants of shift radix systems References

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Three-dimensional symmetric shift radix systems

Cited by 8 publications

References 13 publications

Contractivity of three-dimensional shift radix systems with finiteness property

Contractivity of three-dimensional shift radix systems with finiteness property

Beta-expansions of -adic numbers

Shift Radix Systems - A Survey

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