Endomorphisms of irreducible subshifts of finite type

Coven, Ethan M.; Paul, Michael E.

doi:10.1007/bf01762187

Cited by 118 publications

(68 citation statements)

References 10 publications

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“…So, 82 is fini te-to-one (since, by Proposition 1, right closing maps are conjugate to right resolving maps); thus, any irreducible component 2r (of 2r) with maximal entropy (in 2r) maps (via 82) onto a subshift of 2B with maximal entropy in 2B. By [CP1,Theorem 3.3] this means that 02|s is onto. Now since 82 is right and left closing, it follows that 02|s is constant-to-one (by Proposition 8).…”

Section: Definitionmentioning

confidence: 95%

A note on minimal covers for sofic systems

Boyle¹,

Kitchens²,

Marcus³

1985

Proc. Amer. Math. Soc.

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Abstract.We characterize the sofic systems which have minimal subshift-of-fimtetype covers and derive some consequences. 0. Introduction. For relevant background, see §1. By a factor map, we mean a continuous onto shift commuting map between two symbolic systems. By definition, a sofic system S is a symbolic system which is a factor of a shift of finite type (SFT). In the language of automata theory, sofic systems are regular languages. A minimal cover for S is a SFT, 2^, and a factor map it: 2^ -* S such that any other factor map S (from a SFT onto S) must factor through it: e

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Section: Definitionmentioning

confidence: 95%

A note on minimal covers for sofic systems

Boyle¹,

Kitchens²,

Marcus³

1985

Proc. Amer. Math. Soc.

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“…Begin with HA as in the previous corollary. Obtain a strictly sofic system of the same entropy by identifying the pair of two-blocks [2,3] and [3,2]. This sofic system has a fixed point.…”

Section: Corollarymentioning

confidence: 99%

“…An irreducible subshift of finite type has a unique measure of maximal entropy [9], which is a Markov measure whose matrix has the form P = ¿¿R~XAR (R a diagonal matrix with strictly positive diagonal entries). Any continuous onto finite-to-one factor map between two subshifts of finite type will carry the measure of maximal entropy of one to the measure of maximal entropy of the other [3]. The CurtisHedlund-Lyndon theorem [6] asserts that any continuous factor map between subshifts of finite type is a block map composed with some power of the shift.…”

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confidence: 99%

An invariant for continuous factors of Markov shifts

Kitchens¹

1981

Proc. Amer. Math. Soc.

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Abstract. Let 2ZA and ~ZB be subshifts of finite type with Markov measures (p, P) and (q, Q). It is shown that if there is a continuous onto measure-preserving factor map from ~S.A to 2ZB, then the block of the Jordan form of Q with nonzero eigenvalues is a principal submatrix of the Jordan form of P. If ~LA and 2fi are irreducible with the same topological entropy, then the same relationship holds for the matrices A and B. As a consequence, Jj)(')/£/((')> the rat>° of the zeta functions, is a polynomial. From this it is possible to construct a pair of equalentropy subshifts of finite type that have no common equal-entropy continuous factor of finite type, and a strictly sofic system that cannot have an equal-entropy subshift of finite type as a continuous factor.

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“…Previous research had concentrated on bounded-to-one codes and yielded considerable information about these. We list [1,4,[7][8][9][10][16][17][18][19][20] for examples of material dealing with bounded-to-one codes. The properties of bounded-to-one codes constituted one of our main motivations; it may be useful to compare our results with these.…”

mentioning

confidence: 99%

“…A code <í>: S i T is bounded-to-one if and only if h(S) = h(T) (see [4,16]). Another characterization may be stated as follows.…”

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confidence: 99%

Infinite-to-one codes and Markov measures

Boyle¹,

Tuncel²

1984

Trans. Amer. Math. Soc.

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Abstract. We study the structure of infinite-to-one continuous codes between subshifts of finite type and the behaviour of Markov measures under such codes. We show that if an infinite-to-one code lifts one Markov measure to a Markov measure, then it lifts each Markov measure to uncountably many Markov measures and the fibre over each Markov measure is isomorphic to any other fibre. Calling such a code Markovian, we characterize Markovian codes through pressure. We show that a simple condition on periodic points, necessary for the existence of a code between two subshifts of finite type, is sufficient to construct a Markovian code. Several classes of Markovian codes are studied in the process of proving, illustrating and providing contrast to the main results. A number of examples and counterexamples are given; in particular, we give a continuous code between two Bernoulli shifts such that the defining vector of the image is not a clustering of the defining vector of the domain.0. Introduction. This paper is concerned with the structure of infinite-to-one continuous codes between subshifts of finite type and with the behaviour of Markov measures under such codes. The emergence of notable interest in infinite-to-one codes is recent. This interest was partly fueled by the development of Krieger's marker method [11, 12] and is reflected by the papers [3 and 15]. Previous research had concentrated on bounded-to-one codes and yielded considerable information about these. We list [1,4,[7][8][9][10][16][17][18][19][20] for examples of material dealing with bounded-to-one codes. The properties of bounded-to-one codes constituted one of our main motivations; it may be useful to compare our results with these.After a section on definitions and notation, we will give three examples. The first example concerns infinite-to-one analogues of the resolving codes used in [1, 16 and 17]. The main point about these is that they lift each Markov measure to uncountably many Markov measures. In [15], Marcus, Petersen and Williams construct infinite-to-one codes through which no Markov measure lifts to a Markov measure. Recall that, through a bounded-to-one code, each Markov measure lifts to a unique Markov measure [20]. This lifting process does not alter the memory of the measure when the code is 1-block. However, in the infinite-to-one case there may be strict increases in memory not warranted by the block length of the code: our third (peculiar memory) example is a 1-block code such that no 1-step Markov measure lifts to a 1-step Markov measure, while each 1-step Markov measure has uncountably many 2-step preimages. In addition, this code sends every 1-step Markov

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Endomorphisms of irreducible subshifts of finite type

Cited by 118 publications

References 10 publications

A note on minimal covers for sofic systems

A note on minimal covers for sofic systems

An invariant for continuous factors of Markov shifts

Infinite-to-one codes and Markov measures

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