Flow equivalence of sofic shifts

-ALGEBRAS ASSOCIATED WITH LAMBDA-SYNCHRONIZING SUBSHIFTS AND FLOW EQUIVALENCE

Journal of the Australian Mathematical Society ◽

10.1017/s1446788713000219 ◽

2013 ◽

Vol 95 (2) ◽

pp. 241-265 ◽

Cited By ~ 2

Author(s):

KENGO MATSUMOTO

Keyword(s):

Algebraic Structure ◽

Isomorphism Class ◽

Cartan Subalgebra ◽

Sofic Shift ◽

Sofic Shifts ◽

Flow Equivalence ◽

Graph System

AbstractThe class of $\lambda $-synchronizing subshifts generalizes the class of irreducible sofic shifts. A $\lambda $-synchronizing subshift can be presented by a certain $\lambda $-graph system, called the $\lambda $-synchronizing $\lambda $-graph system. The $\lambda $-synchronizing $\lambda $-graph system of a $\lambda $-synchronizing subshift can be regarded as an analogue of the Fischer cover of an irreducible sofic shift. We will study algebraic structure of the ${C}^{\ast } $-algebra associated with a $\lambda $-synchronizing $\lambda $-graph system and prove that the stable isomorphism class of the ${C}^{\ast } $-algebra with its Cartan subalgebra is invariant under flow equivalence of $\lambda $-synchronizing subshifts.

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A categorical invariant of flow equivalence of shifts

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.74 ◽

2014 ◽

Vol 36 (2) ◽

pp. 470-513 ◽

Cited By ~ 8

Author(s):

ALFREDO COSTA ◽

BENJAMIN STEINBERG

Keyword(s):

Analogous Result ◽

Mild Conditions ◽

Equivalence Of Categories ◽

Natural Equivalence ◽

Sofic Shifts ◽

Flow Equivalence ◽

Flow Invariance ◽

Natural Sense

We prove that the Karoubi envelope of a shift—defined as the Karoubi envelope of the syntactic semigroup of the language of blocks of the shift—is, up to natural equivalence of categories, an invariant of flow equivalence. More precisely, we show that the action of the Karoubi envelope on the Krieger cover of the shift is a flow invariant. An analogous result concerning the Fischer cover of a synchronizing shift is also obtained. From these main results, several flow equivalence invariants—some new and some old—are obtained. We also show that the Karoubi envelope is, in a natural sense, the best possible syntactic invariant of flow equivalence of sofic shifts. Another application concerns the classification of Markov–Dyck and Markov–Motzkin shifts: it is shown that, under mild conditions, two graphs define flow equivalent shifts if and only if they are isomorphic. Shifts with property ($\mathscr{A}$) and their associated semigroups, introduced by Wolfgang Krieger, are interpreted in terms of the Karoubi envelope, yielding a proof of the flow invariance of the associated semigroups in the cases usually considered (a result recently announced by Krieger), and also a proof that property ($\mathscr{A}$) is decidable for sofic shifts.

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Flow equivalence of sofic beta-shifts

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.81 ◽

2016 ◽

Vol 37 (3) ◽

pp. 786-801 ◽

Cited By ~ 2

Author(s):

RUNE JOHANSEN

Keyword(s):

Finite Type ◽

Classification Problem ◽

Flow Classification ◽

Fiber Product ◽

Sofic Shifts ◽

Flow Equivalence

The Fischer, Krieger, and fiber product covers of sofic beta-shifts are constructed and used to show that every strictly sofic beta-shift is 2-sofic. Flow invariants based on the covers are computed, and shown to depend only on a single integer that can easily be determined from the $\unicode[STIX]{x1D6FD}$-expansion of 1. It is shown that any beta-shift is flow equivalent to a beta-shift given by some $1<\unicode[STIX]{x1D6FD}<2$, and concrete constructions lead to further reductions of the flow classification problem. For each sofic beta-shift, there is an action of $\mathbb{Z}/2\mathbb{Z}$ on the edge shift given by the fiber product, and it is shown precisely when there exists a flow equivalence respecting these $\mathbb{Z}/2\mathbb{Z}$-actions. This opens a connection to ongoing efforts to classify general irreducible 2-sofic shifts via flow equivalences of reducible shifts of finite type (SFTs) equipped with $\mathbb{Z}/2\mathbb{Z}$-actions.

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