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 $2\times 2$-matrices with entry sum less than 25, and give examples illustrating to power of the new invariant, i.e., examples where the new invariant can disprove strong shift equivalence whereas the other invariants that we use can not.
We study two problems related to flow equivalence of shift spaces. The first problem, the classification of $S$-gap shifts up to flow equivalence, is partially solved with the establishment of a new invariant for the sofic $S$-gap shifts and a complete classification of the non-sofic $S$-gap shifts. The second problem is an examination of the entropy of shift spaces under flow equivalence. For a wide array of classes of shift spaces with non-zero entropy, it is shown that the entropies achievable while maintaining flow equivalence are dense in $\mathbb R^+$.
We show that Property $(A)$ of subshifts and the semigroup, that is associated to subshifts with Property (A), are invariants of flow equivalence. We show for certain $\mathcal R$-graphs that their isomorphism is implied by the flow equivalence of their $\mathcal R$-graph shifts.
We will study torus actions on Cuntz--Krieger algebras trivially acting on its canonical maximal abelian $C^*$-subalgebras from the view points of continuous orbit equivalence of one-sided topological Markov shifts and flow equivalence of two-sided topological Markov shifts.
We classify certain sofic shifts (the irreducible Point Extension Type, or PET, sofic shifts) up to flow equivalence, using invariants of the canonical Fischer cover. There are two main ingredients: (1) An extension theorem, for extending flow equivalences of subshifts to flow equivalent irreducible shifts of finite type which contain them. (2) The classification of certain constant to one maps from SFTs via algebraic invariants of associated G-SFTs.
In this paper we introduce filtration pairs for isolated invariant sets of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Lastly, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.
We introduce notions of suspension and flow equivalence on one-sided topological Markov shifts, which we call one-sided suspension and one-sided flow equivalence, respectively. We prove that one-sided flow equivalence is equivalent to continuous orbit equivalence on one-sided topological Markov shifts. We also show that the zeta function of the flow on a one-sided suspension is a dynamical zeta function with some potential function and that the set of certain dynamical zeta functions is invariant under one-sided flow equivalence of topological Markov shifts.
We study discrete flow equivalence of two-sided topological Markov shifts by using extended Ruelle algebras. We characterize flow equivalence of two-sided topological Markov shifts in terms of conjugacy of certain actions weighted by ceiling functions of two-dimensional torus on the stabilized extended Ruelle algebras for the Markov shifts.
We introduce a notion of coded equivalence in one-sided topological Markov shifts. The notion is inspired by coding theory. One-sided topological conjugacy implies coded equivalence. We will show that coded equivalence implies continuous orbit equivalence of one-sided topological Markov shifts.
We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.
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 only depend on an single integer easily determined from the $β$-expansion of 1. It is shown that any beta-shift is flow equivalent to a beta-shift given by some $1< β< 2$, 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 i…
The flow equivalence of sofic shifts is examined using results about the structure of the corresponding covers. A canonical cover generalising the left Fischer cover to arbitrary sofic shifts is introduced and used to prove that the left Krieger cover and the past set cover of a sofic shift can be divided into natural layers. These results are used to find the range of a flow invariant and to investigate the ideal structure of the universal C^*-algebras associated to sofic shifts. The right Fischer covers of sofic beta-shifts are constructed, and it is proved that the covering maps are always …
We extend ultragraph shift spaces and the realization of ultragraph C*-algebras as partial crossed products to include ultragraphs with sinks (under a mild condition, called (RFUM2), which allow us to dismiss the use of filters) and we describe the associated transformation groupoid. Using these characterizations we study continuous orbit equivalence of ultragraph shift spaces (via groupoids) and KMS and ground states (via partial crossed products).
We give a new proof of a theorem of Giordano, Putnam and Skau characterizing orbit equivalence of minimal homeomorphisms of the Cantor space in terms of their sets of invariant Borel probability measures. The proof is based on a strengtehning of a theorem of Krieger concerning minimal actions of certain locally finite groups of homeomorphisms, and we also give a new proof of the Giordano--Putnam--Skau characterization of orbit equivalence for these actions.
A $D_{\infty}$-topological Markov chain can be represented by a pair of zero-one square matrices, which is called a flip pair. We introduce the concepts of $D_{\infty}$-strong shift equivalence and $D_{\infty}$-shift equivalence, which are equivalence relations between flip pairs. We investigate the relationships between the existence of a $D_{\infty}$-conjugacy, the existence of a $D_{\infty}$-strong shift equivalence, the existence of a $D_{\infty}$-shift equivalence and the coincidence of the Lind zeta functions.
Previous work showed that every pair of nontrivial Bernoulli shifts over a fixed free group are orbit equivalent. In this paper, we prove that if $G_1,G_2$ are nonabelian free groups of finite rank then every nontrivial Bernoulli shift over $G_1$ is stably orbit equivalent to every nontrivial Bernoulli shift over $G_2$. This answers a question of S. Popa.
We study KMS states for gauge actions with potential functions on Cuntz--Krieger algebras whose underlying one-sided topological Markov shifts are continuous orbit equivalent. As a result, we have a certain relationship between topological entropy of continuous orbit equivalent one-sided topological Markov shifts.
The paper studies algebraic strong shift equivalence of matrices over $n$-variable polynomial rings over a principal ideal domain $D$($n\leq 2$). It is proved that in the case $n=1$, every non-zero matrix over $D[x]$ has a full rank factorization and every non-nilpotent matrix over $D[x]$ is algebraically strong shift equivalent to a nonsingular matrix. In the case $n=2$, an example of non-nilpotent matrix over $\mathbb{R}[x,y,z]=\mathbb{R}[x][y,z]$, which can not be algebraically shift equivalent to a nonsingular matrix, is given.
In this paper we associate an abelian category to a finite directed graph and prove the categories arising from two graphs are equivalent if the incidence matrices of the graphs are shift equivalent. The abelian category is the quotient of the category of graded vector space representations of the quiver obtained by making the graded representations that are the sum of their finite dimensional submodules isomorphic to zero. Actually, the main result in this paper is that the abelian categories are equivalent if the incidence matrices are strong shift equivalent. That result is combined with an…
We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As …
By the work of Brodzki-Niblo-Nowak-Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variations of) these groups are invariants for topologically free actions under continuous orbit equivalence.
We show that the weak and strong forms of the Generalized Spectral Conjecture (GSC) of Boyle and Handelman are equivalent. The GSC asserts that well understood necessary spectral conditions on a square matrix A over a subring S of the reals are sufficient for that matrix to be shift equivalent over S (in the weak form) or strong shift equivalent over S (in the strong form) to a primitive matrix over S. The foundation of this work is the recent result that the group NK_1(S) of algebraic K-theory exactly captures the refinement of shift equivalence over S by strong shift equivalence over S. The …
We study the automorphism group of an infinite minimal shift $(X,σ)$ such that the complexity difference function, $p(n+1)-p(n)$, is bounded. We give some new bounds on $\mbox{Aut}(X,σ)/\langle σ\rangle$ and also study the one-sided case. For a class of Toeplitz shifts, including the class of shifts defined by constant length primitive substitutions with a coincidence and with height one, we show that the two-sided automorphism group is a cyclic group. We next focus on shifts generated by primitive constant length substitutions. For these shifts, we give an algorithm that computes their two-si…
We 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 behavior of the stable conjugacy 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 Marko…
We give sufficient conditions for a positive stochastic matrix to be similar and strong shift equivalent over $\mathbb{R}_+$ 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 $\mathbb{R}_+$ to a positive doubly stochastic matrix. Consequently, the set of nonzero spectra of primitive stochastic matrices over $\mathbb{R}$ with positive trace and the set of nonzero spectra of positive doubly stochastic matrices over $\mathbb{R}$ are identical. We exhibit a class of $2\times 2$ matrices, pai…
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 natur…
The $f$-invariant is a notion of entropy for probability-measure-preserving actions of free groups. We show it is invariant under bounded orbit-equivalence.
To an ergodic, essentially free and measure-preserving action of a non-amenable Baumslag-Solitar group on a standard probability space, a flow is associated. The isomorphism class of the flow is shown to be an invariant of such actions of Baumslag-Solitar groups under weak orbit equivalence. Results on groups which are measure equivalent to Baumslag-Solitar groups are also provided.
The purpose of this article is twofold. On one hand, we reveal the equivalence of shift of finite type between a one-sided shift $X$ and its associated hom tree-shift $\mathcal{T}_{X}$, as well as the equivalence in the sofic shift. On the other hand, we investigate the interrelationship among the comparable mixing properties on tree-shifts as those on multidimensional shift spaces. They include irreducibility, topologically mixing, block gluing, and strong irreducibility, all of which are defined in the spirit of classical multidimensional shift, complete prefix code (CPC), and uniform CPC. I…
In this paper, we will introduce notions of relative version of imprimitivity bimodules and relative version of strong Morita equivalence for pairs of $C^*$-algebras $(\mathcal{A}, \mathcal{D})$ such that $\mathcal{D}$ is a $C^*$-subalgebra of $\mathcal{A}$ with certain conditions. We will then prove that two pairs $(\mathcal{A}_1, \mathcal{D}_1)$ and $(\mathcal{A}_2, \mathcal{D}_2)$ are relatively Morita equivalent if and only if their relative stabilizations are isomorphic. In particularly, for two pairs $(\mathcal{O}_A, \mathcal{D}_A)$ and $(\mathcal{O}_B, \mathcal{D}_B)$ of Cuntz--Krieger …
We introduce the notion of balanced strong shift equivalence between square nonnegative 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 e…
We show that for every metrizable Choquet simplex $K$ and for every group $G$, which is infinite, countable, amenable and residually finite, there exists a Toeplitz $G$-subshift whose set of shift-invariant probability measures is affine homeomorphic to $K$. Furthermore, we get that for every integer $d\geq 1$ and every Toeplitz flow $(X,T)$, there exists a Toeplitz ${\mathbb Z}^d$-subshift which is topologically orbit equivalent to $(X,T)$.
We will study a certain synchronizing property of subshifts called $λ$-synchronization. The $λ$-synchronizing subshifts form a large class of irreducible subshifts containing irreducible sofic shifts. We prove that the $λ$-synchronization is invariant under flow equivalence of subshifts. The $λ$-synchronizing K-groups and the $λ$-synchronizing Bowen-Franks groups are studied and proved to be invariant under flow equivalence of $λ$-synchronizing subshifts. They are new flow equivalence invariants for $λ$-synchronizing subshifts.
Let A,B be square irreducible matrices with entries in {0,1}. We will show that if the one-sided topological Markov shifts (X_A,σ_A) and (X_B,σ_B) are continuously orbit equivalent, then the two-sided topological Markov shifts (\bar X_A,\barσ_A) and (\bar X_B,\barσ_B) are flow equivalent, and hence det(id-A)=det(id-B). As a result, the one-sided topological Markov shifts (X_A,σ_A) and (X_B,σ_B) are continuously orbit equivalent if and only if the Cuntz-Krieger algebras O_A and O_B are isomorphic and det(id-A)=det(id-B).
We introduce a generalization of the notion of approximately proper equivalence relations studied by Renault and with it we build an étale groupoid. Choosing a suitable set of continuous functions to play the role of a potential, we construct a cocycle in that groupoid and discuss the corresponding Radon-Nikodym problem.
Namely, we prove that if $G$ is a countable, discrete group and $f\in M_{n}(\Z(G))$ is invertible on $\ell^{2}(G)^{\oplus n},$ but $f$ is not invertible in $M_{n}(\Z(G))$, then the measure-preserving action of $G$ on $X_{f}$ equipped with the Haar measure is weakly equivalent to a Bernoulli action. We shall in fact prove this weak equivalence in the case that $f$ has a "formal inverse in $\ell^{2}$".
In this paper we study bilipschitz equivalences of germs of holomorphic foliations in $(\mathbb{C}^2,0)$. We prove that the algebraic multiplicity of a singularity is invariant by such equivalences. Moreover, for a large class of singularities, we show that the projective holonomy representation is also a bilipschitz invariant.
For a finite digraph $D$, we define the corresponding subshift of finite type $(X_D, σ_D)$ to be the dynamical system where $X_D$ is the set of all bi-infinite walks through $D$ and $σ_D$ is the shift operator. Two digraphs $D_1$ and $D_2$ are called shift equivalent if there is $m\geq 1$ such that $(X_{D_1}, σ_{D_1}^n)$ and $(X_{D_2}, σ_{D_2}^n)$ are topologically conjugate for all $n\geq m$. They are called strong shift equivalent if this holds for $m=1$. In this paper we introduce a new category which generalises the category of subshifts of finite type and topological conjugacy. Our catego…
We consider probability measure preserving discrete groupoids, group actions and equivalence relations in the context of general probability spaces. We study for these objects the notions of cost, $β$-invariant and some higher-dimensional variants. We also propose various convergence results about $\ell^2$-Betti numbers and rank gradient for sequences of actions, groupoids or equivalence relations under weak finiteness assumptions. In particular we connect the combinatorial cost with the cost of the ultralimit equivalence relations. Finally a relative version of Stuck-Zimmer property is also c…
We will prove that one-sided topological Markov shifts $(X_A,σ_A)$ and $(X_B,σ_B)$ for matrices $A$ and $B$ with entries in $\{0,1\}$ are topologically orbit equivalent if and only if there exists an isomorphism between the Cuntz-Krieger algebras ${\Cal O}_A$ and ${\Cal O}_B$ keeping their commutative $C^*$-subalgerbas $C(X_A)$ and $C(X_B)$. It is also equivalent to the condition that there exists a homeomorphism from $X_A$ to $X_B$ intertwining their topological full groups. We will also study structure of the automorphisms of ${\Cal O}_A$ keeping the commutative $C^*$-algebra $C(X_A)$.
In the early 1990's, Kim and Roush developed path methods for establishing strong shift equivalence (SSE) of positive matrices over a dense subring U of the real numbers R. This paper gives a detailed, unified and generalized presentation of these path methods. New arguments which address arbitrary dense subrings U of R are used to show that for any dense subring U of R, positive matrices over U which have just one nonzero eigenvalue and which are strong shift equivalent over U must be strong shift equivalent over U_+. In addition, we show positive real matrices on a path of shift equivalent p…
The role of generalized Bowen-Franks groups (BF-groups) as topological conjugacy invariants for $\mathbb{T}^{n}$ automorphisms is studied. Using algebraic number theory, a link is established between equality of BF-groups for different automorphisms ($BF$-equivalence) and an identical position in a finite lattice ($\mathcal{L}$-equivalence). Important cases of equivalence of the two conditions are proved. Finally, a topological interpretation of the classical BF-group $\mathbb{Z}% ^{n}/\mathbb{Z}^{n}(I-A)$ in this context is presented.
Given a TQFT in dimension d+1, and an infinite cyclic covering of a closed (d+1)-dimensional manifold M, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams' work in symbolic dynamics. The Turaev-Viro module associated to a TQFT and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary of M has a circle factor and the infinite cyclic cover of the boundary is standard. We define a variant of a TQFT associated to a…
Kengo Matsumoto has introduced lambda-graph systems and strong shift equivalence of lambda-graph systems [Doc.Math.4 (1999), 285-340]. We associate to a subshift of a subshift a lambda-graph system. The strong shift equivalence class of the associated lambda-graph system is an invariant of subsystem equivalence. Wolfgang Krieger and Kengo Matsumoto have introduced the lambda$-entropy of a lambda-graph system and have shown its invariance under strong shift equivalence [Ergod.Th.&Dynam.Sys. 24 (2004) 1155 - 1172]. A separation entropy of a subshift of a subshift is introduced as the lambda-entr…
We show how the fine structure in shift-tail equivalence, appearing in the noncommutative geometry of Cuntz-Krieger algebras developed by the first two authors, has an analogue in a wide range of other Cuntz-Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz-Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz- and Cuntz-Krieger alge…
We initiate the study of correspondences for Smale spaces. Correspondences are shown to provide a notion of a generalized morphism between Smale spaces and are a special case of finite equivalences. Furthermore, for shifts of finite type, a correspondence is related to a matrix which intertwines the adjacency matrices of the shifts. This observation allows us to define an equivalence relation on all Smale spaces which reduces to shift equivalence for shifts of finite type. Several other notions of equivalence are introduced on both correspondences and Smale spaces; a hierarchy between these eq…
We show that the class $\mathscr{B}$, of discrete groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete p.m.p. groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second countable groups, must belong to $\mathscr{B}$. We also introduce a measure-conjugacy invariant called Weak Pinsker entropy and show that, if G is a group in the class $\mathscr{B}$, then Weak Pinsker entropy is an orbit-equivalence…
