Shift Spaces

We define a new class of shift spaces which contains a number of classes of interest, like Sturmian shifts used in discrete geometry. We show that this class is closed under two natural transformations. The first one is called conjugacy and is obtained by sliding block coding. The second one is called the complete bifix decoding, and typically includes codings by non-overlapping blocks of fixed length.

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Sliding block codes between shift spaces over infinite alphabets

Mathematische Nachrichten ◽

10.1002/mana.201500309 ◽

2016 ◽

Vol 289 (17-18) ◽

pp. 2178-2191 ◽

Cited By ~ 7

Author(s):

Daniel Gonçalves ◽

Marcelo Sobottka ◽

Charles Starling

Keyword(s):

Block Codes ◽

Shift Spaces

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Large deviations and fluctuation theorem for selectively decoupled measures on shift spaces

Reviews in Mathematical Physics ◽

10.1142/s0129055x19500363 ◽

2019 ◽

Vol 31 (10) ◽

pp. 1950036 ◽

Cited By ~ 1

Author(s):

Noé Cuneo ◽

Vojkan Jakšić ◽

Claude-Alain Pillet ◽

Armen Shirikyan

Keyword(s):

Multifractal Analysis ◽

Invariant Measures ◽

Large Deviation ◽

Thermodynamic Formalism ◽

Hard Core ◽

Shift Spaces ◽

Large Deviation Principles ◽

Fluctuation Relation ◽

Level 3 ◽

Level 1

We establish the Level-1 and Level-3 Large Deviation Principles (LDPs) for invariant measures on shift spaces over finite alphabets under very general decoupling conditions for which the thermodynamic formalism does not apply. Such decoupling conditions arise naturally in multifractal analysis, in Gibbs states with hard-core interactions, and in the statistics of repeated quantum measurement processes. We also prove the LDP for the entropy production of pairs of such measures and derive the related Fluctuation Relation. The proofs are based on Ruelle–Lanford functions, and the exposition is essentially self-contained.

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Independence entropy of $\mathbb{Z}^{d}$ -shift spaces

Acta Applicandae Mathematicae ◽

10.1007/s10440-013-9819-2 ◽

2013 ◽

Vol 126 (1) ◽

pp. 297-317 ◽

Cited By ~ 5

Author(s):

Erez Louidor ◽

Brian Marcus ◽

Ronnie Pavlov

Keyword(s):

Shift Spaces

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-algebras of labelled graphs III—-theory computations

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.62 ◽

2015 ◽

Vol 37 (2) ◽

pp. 337-368 ◽

Cited By ~ 8

Author(s):

TERESA BATES ◽

TOKE MEIER CARLSEN ◽

DAVID PASK

Keyword(s):

Uniqueness Theorem ◽

Inductive Limit ◽

Important Class ◽

Graph Algebras ◽

Gauge Invariant ◽

Shift Spaces ◽

Graph Algebra ◽

Common Framework ◽

Definition Of ◽

Labelled Graphs

In this paper we give a formula for the$K$-theory of the$C^{\ast }$-algebra of a weakly left-resolving labelled space. This is done by realizing the$C^{\ast }$-algebra of a weakly left-resolving labelled space as the Cuntz–Pimsner algebra of a$C^{\ast }$-correspondence. As a corollary, we obtain a gauge-invariant uniqueness theorem for the$C^{\ast }$-algebra of any weakly left-resolving labelled space. In order to achieve this, we must modify the definition of the$C^{\ast }$-algebra of a weakly left-resolving labelled space. We also establish strong connections between the various classes of$C^{\ast }$-algebras that are associated with shift spaces and labelled graph algebras. Hence, by computing the$K$-theory of a labelled graph algebra, we are providing a common framework for computing the$K$-theory of graph algebras, ultragraph algebras, Exel–Laca algebras, Matsumoto algebras and the$C^{\ast }$-algebras of Carlsen. We provide an inductive limit approach for computing the$K$-groups of an important class of labelled graph algebras, and give examples.

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