Strong Shift Equivalence Research Articles

On strong shift equivalence for row-finite graphs and C*-algebras

On strong shift equivalence for row-finite graphs and C*-algebras

Splittings for $C^*$-correspondences and strong shift equivalence

We present an extension of the notion of in-splits from symbolic dynamics to topological graphs and, more generally, to $C^*$-correspondences. We demonstrate that in-splits provide examples of strong shift equivalences of $C^*$-correspondences. Furthermore, we provide a streamlined treatment of Muhly, Pask, and Tomforde's proof that any strong shift equivalence of regular $C^*$-correspondences induces a (gauge-equivariant) Morita equivalence between Cuntz-Pimsner algebras. For topological graphs, we prove that in-splits induce diagonal-preserving gauge-equivariant $*$-isomorphisms in analogy with the results for Cuntz-Krieger algebras. Additionally, we examine the notion of out-splits for $C^*$-correspondences.

Some Results and Examples on The Relation of Leray Equivalent and (Strong) Shift Equivalent

Using the concept of Leray functor and the way it is used to define the Cohomological Conley index, we define Leray equivalent, Leray shift equivalent, Leray elementary strong shift equivalent and Leray strong shift equivalent. We established their relationship with shift equivalent, strong shift equivalent and elementary strong shift equivalent. Then, we show that in the setting of Artinian and Noetherian module Leray equivalent implies strong shift equivalent.

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

AbstractWe 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.

Subshifts, λ-graph bisystems and C⁎-algebras

We introduce a notion of λ-graph bisystem that consists of a pair (L−,L+) of two labeled Bratteli diagrams L−,L+ satisfying certain compatibility condition for labeling their edges. It is a two-sided extension of λ-graph system, that has been previously introduced by the author. Its matrix presentation is called a symbolic matrix bisystem. We first show that any λ-graph bisystem presents subshifts and conversely any subshift is presented by a λ-graph bisystem, called the canonical λ-graph bisystem for the subshift. We introduce an algebraically defined relation on symbolic matrix bisystems called properly strong shift equivalence and show that two subshifts are topologically conjugate if and only if their canonical symbolic matrix bisystems are properly strong shift equivalent. A λ-graph bisystem (L−,L+) yields a pair of C⁎-algebras written OL−+,OL+− that are first defined as the C⁎-algebras of certain étale groupoids constructed from (L−,L+). We study structure of the C⁎-algebras, and show that they are universal unital unique C⁎-algebras subject to certain operator relations among canonical generators of partial isometries and projections encoded by the structure of the λ-graph bisystem (L−,L+). If a λ-graph bisystem comes from a λ-graph system of a finite directed graph, then the associated subshift is the two-sided topological Markov shift (ΛA,σA) by its transition matrix A of the graph, and the associated C⁎-algebra OL−+ is isomorphic to the Cuntz–Krieger algebra OA, whereas the other C⁎-algebra OL+− is isomorphic to the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z of the commutative C⁎-algebra C(ΛA) of continuous functions on the shift space ΛA of the two-sided topological Markov shift by the automorphism σA⁎ induced by the homeomorphism of the shift σA. This phenomenon shows a duality between Cuntz–Krieger algebra OA and the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z.

Strong topological shift equivalence for dynamical systems

In this paper, we introduce some new equivalence relations for topological dynamical systems named strong topological shift equivalence and topological shift equivalence, which are similar to the strong shift equivalence and shift equivalence for subshifts of finite type. We study the relations between the new equivalences and other equivalences such as topological conjugacy, mutually topological semi-conjugacy and canonical homeomorphism extensions being topologically conjugate. Some properties and examples are shown. In particular, mean topological dimension is an invariant for topological shift equivalence but not for mutually topologically semi-conjugate equivalence. In this topic, linear operators are also considered.

State splitting, strong shift equivalence and stable isomorphism of Cuntz–Krieger algebras

ABSTRACTWe prove that if two nonnegative matrices are strong shift equivalent, the associated stable Cuntz–Krieger algebras with generalized gauge actions are conjugate. The proof is done by a purely functional analytic method and based on constructing imprimitivity bimodule from bipartite directed graphs through strong shift equivalent matrices, so that we may clarify K-theoretic behaviour of the stable isomorphism between the associated stable Cuntz–Krieger algebras. We also examine our machinery for the matrices obtained by state splitting graphs, so that topological conjugacy of the topological Markov shifts is described in terms of some equivalence relation of the Cuntz–Krieger algebras with canonical masas and the gauge actions without stabilization.

Topological Conjugacy of Topological Markov Shifts and Cuntz–Krieger Algebras

For an irreducible non-permutation matrix A , the triplet (\mathcal{O}_A,\mathcal{D}_A,\rho^A) for the Cuntz–Krieger algebra \mathcal{O}_A , its canonical maximal abelian C^\ast -subalgebra \mathcal{D}_A , and its gauge action \rho^A is called the Cuntz–Krieger triplet. We introduce a notion of strong Morita equivalence in the Cuntz–Krieger triplets, and prove that two Cuntz–Krieger triplets (\mathcal{O}_A,\mathcal{D}_A,\rho^A) and (\mathcal{O}_B,\mathcal{D}_B,\rho^B) are strong Morita equivalent if and only if A and B are strong shift equivalent. We also show that the generalized gauge actions on the stabilized Cuntz–Krieger algebras are cocycle conjugate if the underlying matrices are strong shift equivalent. By clarifying K-theoretic behavior of the cocycle conjugacy, we investigate a relationship between cocycle conjugacy of the gauge actions on the stabilized Cuntz–Krieger algebras and topological conjugacy of the underlying topological Markov shifts.

Strong shift equivalence and algebraic K-theory

Abstract For a semiring ℛ \mathcal{R} , the relations of shift equivalence over ℛ \mathcal{R} ( SE- ⁢ ℛ \textup{SE-}\mathcal{R} ) and strong shift equivalence over ℛ \mathcal{R} ( SSE- ⁢ ℛ \textup{SSE-}\mathcal{R} ) are natural equivalence relations on square matrices over ℛ \mathcal{R} , important for symbolic dynamics. When ℛ \mathcal{R} is a ring, we prove that the refinement of SE- ⁢ ℛ \textup{SE-}\mathcal{R} by SSE- ⁢ ℛ \textup{SSE-}\mathcal{R} , in the SE- ⁢ ℛ \textup{SE-}\mathcal{R} class of a matrix A, is classified by the quotient N ⁢ K 1 ⁢ ( ℛ ) / E ⁢ ( A , ℛ ) NK_{1}(\mathcal{R})/E(A,\mathcal{R}) of the algebraic K-theory group N ⁢ K 1 ⁢ ( ℛ ) NK_{1}(\mathcal{R}) . Here, E ⁢ ( A , ℛ ) E(A,\mathcal{R}) is a certain stabilizer group, which we prove must vanish if A is nilpotent or invertible. For this, we first show for any square matrix A over ℛ \mathcal{R} that the refinement of its SE- ⁢ ℛ \textup{SE-}\mathcal{R} class into SSE- ⁢ ℛ \textup{SSE-}\mathcal{R} classes corresponds precisely to the refinement of the GL ⁢ ( ℛ ⁢ [ t ] ) \mathrm{GL}(\mathcal{R}[t]) equivalence class of I - t ⁢ A I-tA into El ⁢ ( ℛ ⁢ [ t ] ) \mathrm{El}(\mathcal{R}[t]) equivalence classes. We then show this refinement is in bijective correspondence with N ⁢ K 1 ⁢ ( ℛ ) / E ⁢ ( A , ℛ ) NK_{1}(\mathcal{R})/E(A,\mathcal{R}) . For a general ring ℛ \mathcal{R} and A invertible, the proof that E ⁢ ( A , ℛ ) E(A,\mathcal{R}) is trivial rests on a theorem of Neeman and Ranicki on the K-theory of noncommutative localizations. For ℛ \mathcal{R} commutative, we show ∪ A E ⁢ ( A , ℛ ) = N ⁢ S ⁢ K 1 ⁢ ( ℛ ) \cup_{A}E(A,\mathcal{R})=NSK_{1}(\mathcal{R}) ; the proof rests on Nenashev’s presentation of K 1 K_{1} of an exact category.

Singular equivalence of finite dimensional algebras with radical square zero

We prove that the Crisp and Gow's quiver operation on a finite quiver Q produces a new quiver Q′ with fewer vertices, such that the finite dimensional algebras kQ/J2 and kQ′/J2 are singularly equivalent. This operation is a general quiver operation which includes as specific examples some operations which arise naturally in symbolic dynamics (e.g., (elementary) strong shift equivalent, (in–out) splitting, source elimination, etc.).

Finite group extensions of shifts of finite type: -theory, Parry and Livšic

This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group$G$, Parry showed how to define a$G$-extension$S_{A}$from a square matrix over$\mathbb{Z}_{+}G$, and classified the extensions up to topological conjugacy by the strong shift equivalence class of$A$over$\mathbb{Z}_{+}G$. Parry asked, in this case, if the dynamical zeta function$\det (I-tA)^{-1}$(which captures the ‘periodic data’ of the extension) would classify the extensions by$G$of a fixed mixing shift of finite type up to a finite number of topological conjugacy classes. When the algebraic$\text{K}$-theory group$\text{NK}_{1}(\mathbb{Z}G)$is non-trivial (e.g. for$G=\mathbb{Z}/n$with$n$not square-free) and the mixing shift of finite type is not just a fixed point, we show that the dynamical zeta function for any such extension is consistent with an infinite number of topological conjugacy classes. Independent of$\text{NK}_{1}(\mathbb{Z}G)$, for every non-trivial abelian$G$we show that there exists a shift of finite type with an infinite family of mixing non-conjugate$G$extensions with the same dynamical zeta function. We define computable complete invariants for the periodic data of the extension for$G$(not necessarily abelian), and extend all the above results to the non-abelian case. There is other work on basic invariants. The constructions require the ‘positive$K$-theory’ setting for positive equivalence of matrices over$\mathbb{Z}G[t]$.

Strong shift equivalence and strong Conley index

ABSTRACTUsing the strong shift equivalence of pointed space maps associated to filtration pairs for isolated invariant sets of a discrete dynamical system we define a strong Conley index independent of the choice of filtration pairs and prove that it is invariant under continuation.

Strong shift equivalence and the Generalized Spectral Conjecture for nonnegative matrices

Given matrices A and B shift equivalent over a dense subring R of R, with A primitive, we show that B is strong shift equivalent over R to a primitive matrix. This result shows that the weak form of the Generalized Spectral Conjecture for primitive matrices implies the strong form. The foundation of this work is the recent result that for any ring R, the group NK1(R) of algebraic K-theory classifies the refinement of shift equivalence by strong shift equivalence for matrices over R.

Strong shift equivalence and positive doubly stochastic matrices

We give sufficient conditions for a positive stochastic matrix to be similar and strong shift equivalent over R+ to a positive doubly stochastic matrix through matrices of the same size. We also prove that every positive stochastic matrix is strong shift equivalent over R+ to a positive doubly stochastic matrix. Consequently, the set of nonzero spectra of primitive stochastic matrices over R with positive trace and the set of nonzero spectra of positive doubly stochastic matrices over R are identical. We exhibit a class of 2×2 matrices, pairwise strong shift equivalent over R+ through 2×2 matrices, for which there is no uniform upper bound on the minimum lag of a strong shift equivalence through matrices of bounded size. In contrast, we show for any n×n primitive matrix of positive trace that the set of positive n×n matrices similar to it contains only finitely many SSE–R+ classes.

Path Methods for Strong Shift Equivalence of Positive Matrices

In the early 1990’s, Kim and Roush developed path methods for establishing strong shift equivalence (SSE) of positive matrices over a dense subring \(\mathcal{U}\) of ℝ. This paper gives a detailed, unified and generalized presentation of these path methods. New arguments which address arbitrary dense subrings \(\mathcal{U}\) of ℝ are used to show that for any dense subring \(\mathcal{U}\) of ℝ, positive matrices over \(\mathcal{U}\) which have just one nonzero eigenvalue and which are strong shift equivalent over \(\mathcal{U}\) must be strong shift equivalent over \(\mathcal{U}_{+}\). In addition, we show matrices on a path of positive shift equivalent real matrices are SSE over ℝ+; positive rational matrices which are SSE over ℝ+ must be SSE over ℚ+; and for any dense subring \(\mathcal{U}\) of ℝ, within the set of positive matrices over \(\mathcal{U}\) which are conjugate over \(\mathcal{U}\) to a given matrix, there are only finitely many SSE-\(\mathcal{U}_{+}\) classes.

On some new invariants for shift equivalence for shifts of finite type

We introduce a new computable invariant for strong shift equivalence of shifts of finite type. The invariant is based on an invariant introduced by Trow, Boyle, and Marcus, but has the advantage of being readily computable. We summarize briefly a large-scale numerical experiment aimed at deciding strong shift equivalence for shifts of finite type given by irreducible 2×2-matrices with entry sum less than 25, and give examples illustrating the power of the new invariant, i.e., examples where the new invariant can disprove strong shift equivalence whereas the other invariants that we use cannot.

A matrix formalism for conjugacies of higher-dimensional shifts of finite type

We develop a natural matrix formalism for state splittings and amalgamations of higher-dimensional subshifts of finite type which extends the common notion of strong shift equivalence of ${\mathbb Z}^+$-matrices. Using the decomposition theorem every topo

On periodic points of $\lambda$-graph systems

In a previous paper (Presentations of subshifts and their topological conjugacy invariants. Doc. Math. 4 (1999), 285?340), the notion of $\lambda$-graph system has been introduced. The $\lambda$-graph systems are generalizations of finite directed labeled graphs. In this paper, we study periodic points of $\lambda$-graph systems. We introduce some invariants for a $\lambda$-graph system $\mathfrak L$ to count the cardinal number of $p$-periodic points of $\mathfrak L$. They are invariant under strong shift equivalence of $\lambda$-graph systems. We then consider the zeta functions of $\lambda$-graph systems, which are also invariant under strong shift equivalence of $\lambda$-graph systems. Some examples are also presented.

Actions of symbolic dynamical systems on C*-algebras

We introduce a notion of C *-symbolic dynamical system, that is a finite family of endomorphisms of a C *-algebra with some conditions. The endomorphisms are indexed by symbols and yield both a nontrivial subshift and a Hilbert C *-bimodule. The associated C *-algebra with the Hilbert C *-bimodule is regarded as a crossed product by the subshift. We also introduce a notion of strong shift equivalence of C *-symbolic dynamical systems. We then prove that if two C *-symbolic dynamical systems are strong shift equivalent, the gauge actions of the crossed product C *-algebras by their subshifts are stably outer conjugate.

Integrality for TQFTs

We discuss ways that the ring of coefficients for a topological quantum field theory (TQFT) can be reduced if one restricts somewhat the allowed cobordisms. When we apply these methods to a TQFT associated to SO(3) at an odd prime p, we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers: ℤ[ζ2p] if p≡−1 (mod 4), and ℤ[ζ4p] if p≡1 (mod 4), where ζk is a primitive kth root of unity. We study the quantum invariants of prime power order simple cyclic covers of 3-manifolds. We define new invariants arising from strong shift equivalence and integrality. Similar results are obtained for some other TQFTs, but the modules are guaranteed only to be projective.