G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory, 3:320-375, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of nite type: shifts whose cover is a nite complement language: there is a nite set F of words over such that cov(X ) F , see Weiss [20] The proper morphisms for shift spaces are given by continuous maps that commute with the shift. By the Curtis Lyndon Hedlund theorem [9], these maps are precisely the global maps of one dimensional cellular automata. For our purposes, a (one dimensional) cellular automaton is simply a local map : The local map naturally extends to a global map that we also denote by : X) i = X i w 1 : X i ) Weiss ....

....in particular to cellular automata, since B( is based on a de Bruijn graph. In fact, permutation automata are a standard way to construct cellular automata that have global maps that are open (in the sense of the usual product topology) and therefore surjective, but fail to be injective, see [9, 18]. The one bit change moves the cover from being trivial, to having maximum possible complexity as a regular language. It is shown in [17] that the minimal Fischer automaton of any factorial, extensible and transitive language L can be described as the uniquely determined strongly connected ....

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G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory, 3:320-375, 1969.

....to its origin as well as for preserving information and energy. The possibilities of Reversible ca (r ca) have been investigated from the 60s: the equivalence between bijectivity and injectivity by Moore and Myhill [11,14] in the 70s: the equivalence of reversibility and bijectivity by Hedlund [5] and Richardson [15] and the decidability of reversibility in dimension 1 of Amoroso and Patt [1] to its undecidability in higher dimension by Kari [6,7] in 1990. The computing power of r ca as well as their simulation powers was particularly investigated in [18] Bennett [2] proved that ....

....1972 [1] but it is undecidable for higher dimension (Kari 1990 [6,7] Whereas for pca and bca, the following lemmas hold in any dimension and states that as far as reversibility is concerned, bca and pca fundamentally dioeer from ca. Recall that bijectivity for ca is equivalent to reversibility [5,15] and that Phi and t work over nite sets so that their bijectivities are decidable. Lemma 2. Morita) A pca is reversible ioe its local function Phi is a permutation. Proof. If Phi is a permutation, then the inverse pca is Gamma Q 2 GammaN S ( Gamma) GammaN ; Phi where GammaN ....

G. A. Hedlund. Endomorphism and automorphism of the shift dynamical system. Mathematical System Theory, 3:320375, 1969.

....collection of finite groups (Theorem 2.3 of [BLR] This kind of result was first given by Curtis, Hedlund and Lyndon who showed that the automorphism group of a full shift contains any finite group and contains a pair of involutions whose product has infinite order (Theorems 6. 13 and 20.1 in [H2]) The paper [H1] contains a survey of their work. The results below are an extension to the case of higher dimensional subshifts of finite type: the discussion before Theorem 2.3 is an analogue of the Curtis Hedlund Lyndon theorem for the full shift on three symbols, and Theorem 2.3 is ....

G.A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Sys. Theory 3 (1969), 320--375.

....temporal, and under # spatial. An easy consequence of the compactness of # is that any such map # must be given by a local rule: there is a neighbourhood size k and a map 2k 1 (#(x) i = f(x i k , x i , x i k ) this is an observation due to Curtis, Lyndon and Hedlund, [12]) A similar definition may be made for automata on one sided shift spaces: define # to be the one sided shift space A N , sum from 0 to only in (1) and define # to be the continuous A to one map defined by (2) for 0 only. Any continuous # commuting map # : # is ....

G.A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3, 320--375 (1969).

....= F (c) continuous (for the product topology) function named the global transition function. Remark We can associate to any shift invariant continuous function F a radius r and a local transition function : Q 2r 1 Q such that F = F where (F (c) i) c(i r) c(i) c(i r) [12]. De nition An Elementary Cellular Automaton (ECA) is a radius 1 two state (usually 0 and 1) onedimensional cellular automaton. Prepared using etds.cls Damage Spreading and sensitivity on Cellular Automata 3 time F F F c 0 F (c 0 ) F 2 (c 0 ) Figure 1. Space time diagram example. For ....

G. A. Hedlund. Endomorphism and automorphism of the shift dynamical system. Mathematical System Theory, 3:320-375, 1969.

....analysis. Note that a cellular automaton whose de Bruijn automaton B(ae) is a permutation automaton is surjective, in fact; the global map is open. Surjectivity of the global map is also equivalent to a strong balance property of B(ae) every word has to have the same multiplicity in B(ae) see [4] and [17] In particular, the number of transitions in B(ae) labeled by any particular symbol in the alphabet is the same as for any other symbol. It was suggested by Langton to use the simple numerical parameter (ae) k w Gamma k 0 k w as a measure for the complexity of a CA, see [14] ....

....automata as for finite state machines and their acceptance languages. Thus, a CA ae is a permutation automaton if B(ae) is a permutation automaton, ae) stands for (L(ae) and so forth. Permutation CAs are trivially surjective and even open: their global maps are exactly k w Gamma1 to 1, see [4]. The reference also shows that a cellular automaton ae is a permutation automaton iff there is a bilaterally transitive configuration with k w Gamma1 preimages under ae 1 . Because of the regularity of the de Bruijn graphs underlying the semiautomata B(ae) it is easy to count permutation ....

G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory, 3:320--375, 1969. 25

.... in cellular automata proliferated[3,61,54,46 48,90,35] In spite of that work, however, for many #4 Unbeknownst to those authors, systems that are in essence one dimensional cellular automata had already been studied in an abstract mathematical context by Hedlund and associates as early as 1963[30,31]; both Richardson s results on invertibility (x4.3) and Patt s search for ica (x5.3) had been anticipated by Hedlund s school. T. Toffoli, N. H. Margolus = Invertible Cellular Automata 4 years the most interesting ica actually exhibited remained an extremely simple minded one (the longest orbit ....

....support the simulation of ica which are likely to constitute a major portion of its fare. T. Toffoli, N. H. Margolus = Invertible Cellular Automata 5 2.4 Terminology The present section complements with precise definitions and notation the informal terminology introduced in x2.1 2.2. Refer to [31,72] for more abstract, but equivalent, definitions given in terms of continuity in the Cantor set topology. Space. Let S = Z n denote the Abelian group of translations of an n dimensional lattice I onto itself. It will be convenient to call the elements of S displacements. The sum and the ....

Hedlund, G. A., "Endomorphismand Automorphism of the Shift Dynamical System," Math. Syst. Theory 3 (1969), 51--59.

....of nite type: shifts whose cover is a nite complement language: there is a nite set F of words over such that cov(X ) F , see Weiss [20] The proper morphisms for shift spaces are given by continuous maps that commute with the shift. By the Curtis Lyndon Hedlund theorem [9], these maps are precisely the global maps of one dimensional cellular automata. For our purposes, a (one dimensional) cellular automaton is simply a local map : w . The local map naturally extends to a global map 1 1 that we also denote by : X) i = X i w 1 : X i ) ....

....in particular to cellular automata, since B( is based on a de Bruijn graph. In fact, permutation automata are a standard way to construct cellular automata that have global maps that are open (in the sense of the usual product topology) and therefore surjective, but fail to be injective, see [9, 18]. The one bit change moves the cover from being trivial, to having maximum possible complexity as a regular language. It is shown in [17] that the minimal Fischer automaton of any factorial, extensible and transitive language L can be described as the uniquely determined strongly connected ....

[Article contains additional citation context not shown here]

G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory, 3:320-375, 1969.

....[Kr1] which are reviewed in Section 3. In Section 6 we prove that if F is assumed to be a shift of nite type and N is a power of a prime p, then the number J above satis es J p 2 . The proof uses measure multipliers (reviewed in Section 5) developed in [B] to generalize Welch s theory [H] of compatible extension numbers to shifts of nite type. In Section 7, we make three conjectures about the possible dynamics of an expansive automorphism of SN . Two of these were originally introduced by M. Nasu in the form of questions, to which our results give partial answers. In Section 8 ....

....the rank of Bilat(F ) is 1. It follows from Proposition 3.2 that F is an SFT shift equivalent to a full shift. Remark 4.3. Let J be the integer such that F is shift equivalent to J . Then SN may be viewed as an N to 1 endomorphism of some power of J , and it follows from Welch s theorem ([H], Theorem 14.9) or its generalizations ( B] T1] T2] that every prime dividing N must also divide J . But to show the same primes divide N and J requires more, since (for example) the rule ( x)n = 3xn xn 1 (mod 6) de nes a 2 to 1 but not positively expansive endomorphism of 6 . ....

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G.A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math.Systems Th. 3 (1969), 320-375.

....results we need to reformulate surjectivity as a property on the local rule: balance. A CA with radius r is k balanced if 8y 2 S 2r(k 1) 1 ; n x 2 S 2rk 1 j f(x) y o = S 2r : The following proposition states the equivalence between balance and surjectivity. Proposition 1 (Hedlund [12]) A 1 D CA is surjective i it is k balanced for all k 2 N. 4 Chaos has no edges Classifying dynamical behavior using a parameter means to have a one toone correspondence between intervals and types of behavior. In practical situations one may also require the parameter to be e ective, i.e. ....

G. A. Hedlund. Endomorphism and automorphism of the shift dynamical system. Math. Sys. Theory, 3:320-375, 1969. 16

....e [a] Theta b m j m2M where, 8m; b m = a e:m , where : is the monoid operator ( for M = Z D Theta N E , etc. A cellular automaton (CA) is a continuous self map F : A M Gamma A M which commutes with all shifts: for any e 2 M , F ffi oe e = oe e ffi F. Hedlund [2] proved that any such map is determined by a local function f : A U Gamma A, where U ae M is some finite set (thought of as a neighbourhood around the identity in M ) so that, for any a = Theta am j m2M 2 A M , with F(a) Theta b m j m2M 2 A M , we have: 8m 2 M ; b ....

....p Gamma1 Gamma Gamma . Proof: B has upper density ffi, so there is some sequence fn k g 1 k=0 such that, Card [B [0; n k ] n k Gamma Gamma Gamma Gamma k 1 Gamma ffi: Find K so that, for k K, Card [B [0: n k ] n k ffi 2 . Then choose M large enough to satisfy [1] and [2], and such that p M Gamma1 n k p M . Thus, Card Theta B Theta 0; p M Card [B [0; n k ] ffin k 2 ffip M Gamma1 2 ffip M 2p : 10) Also, by Claim 1, let M be large enough so that there is a subset G M w (ffl) ae [0: p) M so that Card Theta G M w ....

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G. Hedlund. Endomorphisms and automorphisms of the shift dynamical systems. Mathematical System Theory, 3:320--375, 1969.

....T : X 1 ; S 1 ) X 2 ; T 2 ) is countable to one 12 almost everywhere. However, our result is for d = 2, does not insist that T is onto, and also does not require that T is countable to one almost everywhere. Partly as pre requisites for the above work, we generalise some of Hedlund s results [21] for one dimensional cellular automata (continuous maps of( Omega ; oe) to continuous maps of ( Sigma; oe) where ( Sigma; oe) is a two dimensional subshift of finite type (we use the term two dimensional subcellular automata to refer to such maps) We also show (for d = 2) that any corner ....

....62 the alphabet size of the original system) that can be represented by a k Theta k matrix A of zeroes and ones. If A is irreducible (for all 1 i; j k there exists n 2 N such that the (i; j)th entry of A n is non zero) we say that the matrix subshift of finite type is irreducible. Hedlund [21] showed (the result was joint work of his, with Curtis and Linden) that, for d = 1 and given k 2 N, the set of cellular automata maps is equivalent to the set of maps f1 of( Omega ; oe) determined by finite sets E ae Z and maps f : S E S given by (f 1 (x) n = f(E n (x) f(E (oe n (x) ....

G.A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Sys. Th. 3 320--375 (1969). 87

.... k) mod n) c 2 A n ; 0 i n: Let f; f : A 2k 1 A; be a local rule, and m the cardinality of A. Note that if c 2 A n is a periodic configuration for F n , then c 1 is a periodic configuration for F . We now give the definitions of permutive and additive local rule. Definition 1 ([16]) f is permutive in x i , Gammak i k; if and only if, for any given sequence x Gammak ; x i Gamma1 ; x i 1 ; x k 2 A 2k ; we have ff(x Gammak ; x i Gamma1 ; x i ; x i 1 ; x k ) x i 2 Ag = A: Definition 2 f is leftmost [rightmost] permutive if and only if ....

....of leftmost and or rightmost permutive CA which includes both additive and non additive CA. In [13] it has been shown that there exist transitive CA which are neither leftmost nor rightmost permutive. The following lemmas allow us to prove that additive CA have dense periodic orbits. 8 Lemma 2 ([16]) Let b 2 A l be a finite configuration of length l defined on A = f0; 1; m Gamma 1g: Let N(n) be the number of configurations of length n which contain b. Then lim n 1 N(n) m n = 1: Let f be an additive local rule defined by f(x Gammak ; x k ) 0 k X j= Gammak ....

G. A. Hedlund, Endomorphism and Automorphism of the Shift Dynamical System. Mathematical System Theory 3(4), 320-375, 1970.

....f u ranging over Z. However if A is nonabelian, the term linear is rather misleading; some authors have taken to calling these automata group automata for this reason. Intuitively, linear automata should do a good job of generating randomness . For example, these automata are permutative [7] (suggesting that any local perturbation of a configuration is afforded the maximum opportunity of spreading outwards) and have been shown [6] 2] in many cases to be chaotic in the sense of Devaney [3] Also, linear cellular automata are automorphisms of compact groups; when endowed with the ....

G. Hedlund. Endomorphisms and automorphisms of the shift dynamical systems. Mathematical System Theory, 3:320--375, 1969.

....OE is then the function Phi : A Z D Gamma A Z D sending Theta an j n2Z D 7 Theta OE n (aU n) j n2Z D . OE is called the local transformation rule for Phi. Cellular automata were first investigated by Von Neumann [25] and Ulam [24] and later extensively studied by Hedlund [6], Wolfram [27] and others; more recent surveys are [23] 5] 12] 3] Any cellular automaton on Z D can be represented by a subshift of finite type on Z D Theta Z. Simply define e U : U Theta f0g) t 8 : 0; 0; 0 z D ; 1) 9 = and then set f W : n a 2 A e ....

G. Hedlund. Endomorphisms and automorphisms of the shift dynamical systems. Mathematical System Theory, 3:320--375, 1969.

....# denote the usual shift map #(s 0 s 1 s 2 . s 1 s 2 . An important question in symbolic dynamics concerns the group of automorphisms of the shift, i.e. maps # :# d # # d that commute with the shift map. For one sided shift maps, this group is well understood thanks to work of Hedlund [6], Boyle, Franks, and Kitchens [3] and Ashley [1] The automorphism group for the 2 shift is simple: There is only one non trivial element, namely the automorphism that interchanges the two symbols 0 and 1. For the d shift, the group is infinitely generated with a rich algebraic structure. ....

Hedlund, G. Endomorphisms and Automorphisms of the Shift Dynamical System. Math. Syst. Theory 3 (1969).

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G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory, 3:320-375, 1969.

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G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory, 3:320-375, 1969.

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G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Mathematical Systems Theory, 3(4):320-375, 1969.

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G. A. Hedlund, Endomorphisms and Automorphisms of the Shifts Dynamical System, Math. Systems Th. 3, 320-375 (1969).

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G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Th. 3 (1969), 320-375.

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G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory, 3:320-375, 1969.

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G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory, 3:320-375, 1969.

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Hedlund, G. Endomorphisms and Automorphisms of the Shift Dynamical System. Math. Syst. Theory 3 (1969).

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G. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory 3 (1969), 320-375.

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