scholarly journals Flow equivalence of sofic beta-shifts

2016 ◽  
Vol 37 (3) ◽  
pp. 786-801 ◽  
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.

scholarly journals -ALGEBRAS ASSOCIATED WITH LAMBDA-SYNCHRONIZING SUBSHIFTS AND FLOW EQUIVALENCE

2013 ◽  
Vol 95 (2) ◽  
pp. 241-265 ◽  
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.

Flow equivalence of subshifts of finite type

1984 ◽  
Vol 4 (1) ◽  
pp. 53-66 ◽  
Author(s):  
John Franks
Keyword(s):  
Finite Type ◽  
Suspension Flows ◽  
Flow Equivalence ◽  
Complete Set ◽  
Subshifts Of Finite Type

AbstractA complete set of computable invariants is given for deciding whether two irreducible subshifts of finite type have topologically equivalent suspension flows.

scholarly journals Normal amenable subgroups of the automorphism group of sofic shifts

2020 ◽  
pp. 1-14
Author(s):  
KITTY YANG
Keyword(s):  
Automorphism Group ◽  
Finite Type ◽  
Higher Dimensions ◽  
Two Dimensional ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Sofic Shifts ◽  
Amenable Subgroup

Let $(X,\unicode[STIX]{x1D70E})$ be a transitive sofic shift and let $\operatorname{Aut}(X)$ denote its automorphism group. We generalize a result of Frisch, Schlank, and Tamuz to show that any normal amenable subgroup of $\operatorname{Aut}(X)$ must be contained in the subgroup generated by the shift. We also show that the result does not extend to higher dimensions by giving an example of a two-dimensional mixing shift of finite type due to Hochman whose automorphism group is amenable and not generated by the shift maps.

scholarly journals A categorical invariant of flow equivalence of shifts

2014 ◽  
Vol 36 (2) ◽  
pp. 470-513 ◽  
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.

Lattice invariants for sofic shifts

1991 ◽  
Vol 11 (4) ◽  
pp. 787-801 ◽  
Author(s):  
Susan Williams
Keyword(s):  
Finite Type ◽  
Necessary Conditions ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Dimension Group ◽  
Sofic Shifts ◽  
Factor Map ◽  
Factor Maps

AbstractTo a factor map φ from an irreducible shift of finite type ΣAto a sofic shiftS, we associate a subgroup of the dimension group (GA, Â) which is an invariant of eventual conjugacy for φ. This invariant yields new necessary conditions for the existence of factor maps between equal entropy sofic shifts.

scholarly journals Decompositions of factor codes and embeddings between shift spaces with unequal entropies

2012 ◽  
Vol 33 (1) ◽  
pp. 144-157
Author(s):  
SOONJO HONG ◽  
UIJIN JUNG ◽  
IN-JE LEE
Keyword(s):  
Finite Type ◽  
Shift Spaces ◽  
Sofic Shift ◽  
Shift Space ◽  
Sofic Shifts ◽  
The Given ◽  
Factor Code

AbstractGiven a factor code between sofic shifts X and Y, there is a family of decompositions of the original code into factor codes such that the entropies of the intermediate subshifts arising from the decompositions are dense in the interval from the entropy of Y to that of X. Furthermore, if X is of finite type, we can choose those intermediate subshifts as shifts of finite type. In the second part of the paper, given an embedding from a shift space to an irreducible sofic shift, we characterize the set of the entropies of the intermediate subshifts arising from the decompositions of the given embedding into embeddings.

Flow equivalence of reducible shifts of finite type

1994 ◽  
Vol 14 (4) ◽  
pp. 695-720 ◽  
Author(s):  
Danrung Huang
Keyword(s):  
Finite Type ◽  
Irreducible Components ◽  
Flow Equivalence

AbstractUsing an invariant of Cuntz, we classify reducible shifts of finite type with two irreducible components up to flow equivalence.

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