scholarly journals Balanced strong shift equivalence, balanced in-splits, and eventual conjugacy

2020 ◽  
pp. 1-21
Author(s):  
KEVIN AGUYAR BRIX
Keyword(s):  
Finite Type ◽  
Negative Integer ◽  
The Other ◽  
Finite Graphs ◽  
Strong Shift ◽  
Shift Equivalence ◽  
Adjacency Matrices ◽  
Strong Shift Equivalence ◽  
The One

Abstract We introduce the notion of balanced strong shift equivalence between square non-negative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits, the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.

scholarly journals CUNTZ–KRIEGER ALGEBRAS AND ONE-SIDED CONJUGACY OF SHIFTS OF FINITE TYPE AND THEIR GROUPOIDS

2019 ◽  
Vol 109 (3) ◽  
pp. 289-298
Author(s):  
KEVIN AGUYAR BRIX ◽  
TOKE MEIER CARLSEN
Keyword(s):  
Conjugacy Class ◽  
Finite Type ◽  
Negative Answer ◽  
The Other ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Positive Map ◽  
Shift Of Finite Type ◽  
Abelian Subalgebra ◽  
The One

AbstractA one-sided shift of finite type $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines on the one hand a Cuntz–Krieger algebra ${\mathcal{O}}_{A}$ with a distinguished abelian subalgebra ${\mathcal{D}}_{A}$ and a certain completely positive map $\unicode[STIX]{x1D70F}_{A}$ on ${\mathcal{O}}_{A}$. On the other hand, $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines a groupoid ${\mathcal{G}}_{A}$ together with a certain homomorphism $\unicode[STIX]{x1D716}_{A}$ on ${\mathcal{G}}_{A}$. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of $\mathsf{X}_{A}$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

scholarly journals Strong shift equivalence of 2 × 2 matrices of non-negative integers

1983 ◽  
Vol 3 (4) ◽  
pp. 501-508 ◽  
Author(s):  
Kirby A. Baker
Keyword(s):  
Markov Chains ◽  
Sufficient Conditions ◽  
Topological Isomorphism ◽  
Strong Shift ◽  
Shift Equivalence ◽  
Integral Matrices ◽  
Strong Shift Equivalence

AbstractThe concept of strong shift equivalence of square non-negative integral matrices has been used by R. F. Williams to characterize topological isomorphism of the associated topological Markov chains. However, not much has been known about sufficient conditions for strong shift equivalence even for 2×2 matrices (other than those of unit determinant). The main theorem of this paper is: If A and B are positive 2×2 integral matrices of non-negative determinant and are similar over the integers, then A and B are strongly shift equivalent.

Strong shift equivalence and strong Conley index

2015 ◽  
Vol 31 (3) ◽  
pp. 280-292
Author(s):  
Sainkupar Marwein Mawiong ◽  
Himadri Kumar Mukerjee
Keyword(s):  
Conley Index ◽  
Strong Shift ◽  
Shift Equivalence ◽  
Strong Shift Equivalence

scholarly journals INVARIANT HYPERSURFACES

2020 ◽  
pp. 1-27
Author(s):  
Jason Bell ◽  
Rahim Moosa ◽  
Adam Topaz
Keyword(s):  
Rational Function ◽  
Finite Type ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Inverse Image ◽  
The Other ◽  
Partial Result ◽  
Rational Maps ◽  
Special Cases ◽  
The One

The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$ -varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose $\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$ are dominant rational maps from an (possibly nonreduced) irreducible scheme $Z$ of finite type to an algebraic variety $X$ , with the property that there are infinitely many hypersurfaces on  $X$ whose scheme-theoretic inverse images under $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D719}_{2}$ agree. Then there is a nonconstant rational function $g$ on $X$ such that $g\unicode[STIX]{x1D719}_{1}=g\unicode[STIX]{x1D719}_{2}$ . In the case where $Z$ is also reduced, the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou–Hrushovski theorem to generalised algebraic ${\mathcal{D}}$ -varieties and of Cantat’s theorem to self-correspondences.

scholarly journals Path Methods for Strong Shift Equivalence of Positive Matrices

2013 ◽  
Vol 126 (1) ◽  
pp. 65-115 ◽  
Author(s):  
Mike Boyle ◽  
K. H. Kim ◽  
F. W. Roush
Keyword(s):  
Strong Shift ◽  
Shift Equivalence ◽  
Positive Matrices ◽  
Strong Shift Equivalence

scholarly journals Strong shift equivalence of C*-correspondences

2008 ◽  
Vol 167 (1) ◽  
pp. 315-346 ◽  
Author(s):  
Paul S. Muhly ◽  
David Pask ◽  
Mark Tomforde
Keyword(s):  
Strong Shift ◽  
Shift Equivalence ◽  
Strong Shift Equivalence

scholarly journals On strong shift equivalence over a Boolean semiring

1986 ◽  
Vol 6 (1) ◽  
pp. 81-97 ◽  
Author(s):  
Ki Hang Kim ◽  
Fred W. Roush
Keyword(s):  
Equivalence Relation ◽  
Equivalence Classes ◽  
Directed Edge ◽  
Strong Shift ◽  
Shift Equivalence ◽  
Edge Graph ◽  
Boolean Semiring ◽  
Strong Shift Equivalence

AbstractShift equivalence is the relation between A, B that there exists S, R, n > 0 with RA = BR, AS = SB, SR = An, RS = Bn. Strong shift equivalence is the equivalence relation generated by these equations with n = 1. We prove that for many Boolean matrices strong shift equivalence is characterized by shift equivalence and a trace condition. However, we also show that if A is strongly shift equivalent to B, then there exists a homomorphism from an iterated directed edge graph of A to the graph of B preserving the traces of powers. This yields results on colourings of iterated directed edge graphs and might distinguish new strong equivalence classes.

Sign in / Sign up

Export Citation Format

Share Document