A counterexample to the unimodular conjecture on finitely generated dimension groups

Riedel, Norbert

doi:10.1090/s0002-9939-1981-0619970-x

Cited by 7 publications

(6 citation statements)

References 3 publications

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“…where k ≥ 1 and all M n are unimodular matrices. It was proved to hold when G is simple and has one state Riedel (1981a) but disproved in the general case Riedel (1981b).…”

Section: Dimension Groupsmentioning

confidence: 99%

Dimension Groups and Dynamical Systems

Durand¹,

Perrin²

2020

Preprint

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The text follows, at least in an initial version, closely the notes of Bernard Host Host (1995), trying to develop more explicitly some arguments. We have also chosen to follow a slightly different presentation, not assuming systematically that the dynamical systems are minimal. We wish to thank Brian Marcus and Mike Boyle for fruitful discussions during the preparation of this text. We are also grateful to Paulina Cecchi, Francesco Dolce, Pavel Heller, Christophe Reutenauer and Revekka Kyriakoglou for reading the manuscript and founding many errors. Special thanks are due to Soren Eilers for reading closely Chapter 10 and to Christian for his careful reading of most chapters.Proposition 2.1.4 A topological dynamical system is minimal if and only if the positive orbit of every point is dense in X. Example 2.5.1 The Fibonacci shift (see Example 2.4.1) is Sturmian (see Exercise 2.32).As one might expect in view of its minimal word complexity, any Sturmian shift is minimal, moreover it is uniquely ergodic.

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“…where k ≥ 1 and all M n are unimodular matrices. It was proved to hold when G is simple and has one state Riedel (1981a) but disproved in the general case Riedel (1981b).…”

Section: Dimension Groupsmentioning

confidence: 99%

Dimension Groups and Dynamical Systems

Durand¹,

Perrin²

2020

Preprint

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“…This was conjectured in different terms in [ES79]. It was shown to be true when the dimension group has a unique trace [Rie81a] (or, equivalently, when all minimal systems in this class are uniquely ergodic) but shown to be false in general [Rie81b]. In the same spirit, one may ask if the strong orbit equivalence class of any primitive unimodular proper S-adic subshift contains a dendric subshift.…”

Section: Infinitesimals and Balancednessmentioning

confidence: 99%

“…According to [Rie81b] (see also Section 6.5), not all strong orbit equivalence classes represented by dimension groups of the type (1) in Theorem 4.5 contain primitive unimodular proper S-adic subshifts. The description of the dynamical dimension group in Theorem 4.5 is not precise enough to explain the restrictions that occur for instance for the measures, so that a complete characterization of the dynamical dimension groups of primitive unimodular proper S-adic subshifts is still missing.…”

Section: Questions and Further Workmentioning

confidence: 99%

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On The Dimension Group of Unimodular S-Adic Subshifts

Berthe,

Bernales,

Durand

et al. 2019

Preprint

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Dimension groups are complete invariants of strong orbit equivalence for minimal Cantor systems. This paper studies a natural family of minimal Cantor systems having a finitely generated dimension group, namely the primitive unimodular proper S-adic subshifts. They are generated by iterating sequences of substitutions. Proper substitutions are such that the images of letters start with a same letter, and similarly end with a same letter. This family includes various classes of subshifts such as Brun subshifts or dendric subshifts, that in turn include Arnoux-Rauzy subshifts and natural coding of interval exchange transformations. We compute their dimension group and investigate the relation between the triviality of the infinitesimal subgroup and rational independence of letter measures. We also introduce the notion of balanced functions and provide a topological characterization of balancedness for primitive unimodular proper S-adic subshifts.

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“…After proving his result for the case in which there is a unique state with image rank greater than one, Riedel showed in [12] that this cannot be extended to simple dimension groups with more than one state; specifically, for any n > 1 there is a simple dimension group with n + 1 generators and n extreme states that is not ultrasimplicial. There is no overlap between the counterexamples constructed in [12] and the groups discussed in the present work, because in the present work there is only one state.…”

Section: Introductionmentioning

confidence: 99%

The ultrasimplicial property for simple dimension groups with unique state, the image of which has rank one

Maloney

2014

Preprint

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A counterexample to the unimodular conjecture on finitely generated dimension groups

Abstract: Abstract. We give a series of examples of simple finitely generated dimension groups which cannot be obtained as the inductive limit of a systemwhere each A" is a unimodular matrix whose entries are nonnegative integers.

Cited by 7 publications

References 3 publications

Dimension Groups and Dynamical Systems

Dimension Groups and Dynamical Systems

On The Dimension Group of Unimodular S-Adic Subshifts

The ultrasimplicial property for simple dimension groups with unique state, the image of which has rank one

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