Constant-to-one and onto global maps of homomorphisms between strongly connected graphs

Nasu, Masakazu

doi:10.1017/s0143385700002042

Cited by 61 publications

(43 citation statements)

References 18 publications

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“…. , k} and the shift a Ai can be represented as (2)) be chosen continuous and in fact depend on only finitely many coordinates (see [8]). It follows then that the solution h to (3) (if it exists) must depend on only finitely many coordinates as well, and in fact (3) Proof.…”

Section: Introductionmentioning

confidence: 99%

Almost topological classification of finite-to-one factor maps between shifts of finite type

Adler

Kitchens

Marcus³

1985

Ergod. Th. Dynam. Sys.

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Abstract. We classify finite-to-one factor maps between shifts of finite type up to almost topological conjugacy. IntroductionIn [2], shifts of finite type (SFT) were classified up to almost topological conjugacy. The purpose of this paper is to classify finite-to-one continuous factor maps between shifts of finite type. We consider two equivalence relations. The first is very strong: two maps are equivalent if they have the same range shift and they differ by a special kind of almost continuous change of variables in the domain; for a fixed range shift and a fixed generic cardinality of the fibre of the factor maps (a natural invariant), we get infinitely many equivalence classes. The second (more natural) equivalence relation allows the same kind of change of variables in the range (as well as in the domain) and gives only finitely many equivalence classes (for a fixed entropy class and generic cardinality of the fibre). For the latter classification, we reduce the problem to a group action problem which was solved by us in [1]. Our results are completely analogous to and were inspired by those of D. Rudolph [10] in the measure-theoretic category (i.e. classification of finite-to-one factor maps between Bernoulli shifts). Our work is basically a more concrete version of Rudolph's work. In particular, let T be a Bernoulli shift whose measure-theoretic entropy (log (A)) is the same as the topological entropy of an aperiodic SFT; then for any finite-to-one factor map of T, we construct (theorem 4.3) an equivalent (in the measure-theoretic sense) factor map w:2 A ->S B , where 2 B is an arbitrary aperiodic SFT of entropy log (A) and S A is some aperiodic SFT of entropy log (A). (In general, 2 A cannot be chosen arbitrarily.) It would perhaps have been most natural for us to classify our factor maps with respect to continuous changes of variables (i.e. topological conjugacy) since, after all, our factor maps are continuous themselves. But the simplest case of this, when the factor maps involved are identity maps, reduces to the classical topological conjugacy problem for shifts of finite type-which is not yet solved satisfactorily.For background, we basically refer to [2]. We use the notation 2 A for an SFT defined by 0-1 transition matrix A, cr A for the shift, and L A for the alphabet (or

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Section: Introductionmentioning

confidence: 99%

Almost topological classification of finite-to-one factor maps between shifts of finite type

Adler

Kitchens

Marcus³

1985

Ergod. Th. Dynam. Sys.

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“…The elementary definition we shall give has appeared in many places in the scientific literature, most notably in symbolic dynamics (left/right covers [22], regular homomorphisms [25]) and spectral graph theory (divisors [28] and semicovers [14]). However, a graph morphism has an immediate interpretation as a functor between free categories, and in that case Grothendieck's oldest notion of fibration [10] reduces exactly to the elementary definition we shall use.…”

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confidence: 99%

Graph fibrations, graph isomorphism, and PageRank

Boldi

Lonati

Santini

et al. 2006

RAIRO-Theor. Inf. Appl.

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Abstract. PageRank is a ranking method that assigns scores to web pages using the limit distribution of a random walk on the web graph. A fibration of graphs is a morphism that is a local isomorphism of inneighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. We show that a deep connection relates fibrations and Markov chains with restart, a particular kind of Markov chains that include the PageRank one as a special case. This fact provides constraints on the values that PageRank can assume. Using our results, we show that a recently defined class of graphs that admit a polynomial-time isomorphism algorithm based on the computation of PageRank is really a subclass of fibration-prime graphs, which possess simple, entirely discrete polynomial-time isomorphism algorithms based on classical techniques for graph isomorphism. We discuss efficiency issues in the implementation of such algorithms for the particular case of web graphs, in which O(n) space occupancy (where n is the number of nodes) may be acceptable, but O(m) is not (where m is the number of arcs).Mathematics Subject Classification. 05C50, 05C85, 05C60, 94C15, 60J10, 15A51.

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“…Then, since q is a biresolving graph-homomorphism with q V N -to-one, it follows that l 2 and r 2 are left and right eigenvectors of M Γ corresponding to J and l 2 r 2 = N l 0 r 0 (see [N1,Proposition 2.3]). Since l 1 and l 2 are eigenvectors of the same irreducible nonnegative matrix corresponding to its spectral radius, we have Proposition 6.1 for the case that (X G , σ G ) is a full shift (i.e., G is a one-vertex graph) was already known: (2) is found in [SA] and [N2], and (3) is contained in the result in [C] A → G A ) be the textile system such that p * and q * are given by p * (a 1 a 2 ) = a 1 and q * (a 1 a 2 ) = f (a 1 a 2 ) with a 1 , a 2 ∈ A.…”

Section: Bipermutive Endomorphisms Of Mixing Topological Markov Shiftsmentioning

confidence: 99%

The dynamics of expansive invertible onesided cellular automata

Nasu

2002

Trans. Amer. Math. Soc.

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Constant-to-one and onto global maps of homomorphisms between strongly connected graphs

Cited by 61 publications

References 18 publications

Almost topological classification of finite-to-one factor maps between shifts of finite type

Almost topological classification of finite-to-one factor maps between shifts of finite type

Graph fibrations, graph isomorphism, and PageRank

The dynamics of expansive invertible onesided cellular automata

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