D. Lind, B. Marcus, Symbolic Dynamics and Coding, Cambridge University Press, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to be especially significant, leading to breakthroughs in both the theory and design of constrained codes. An interesting account of this development and its impact on the design of recording codes for magnetic storage is given in [2] A very comprehensive mathematical treatment may be found in [138]. C. Properties of Graph Labelings In order to state the coding theorems, as well as for purposes of encoder construction, it will be important to consider labelings with special properties. We say that a labeled graph is deterministic if, at each state, the outgoing edges have distinct labels. ....

....D. Finite Type and Almost Finite Type Constraints There are some special classes of constraints, called finitetype and almost finite type, that play an important role in the theory and construction of constrained codes. A constrained system is finite type (a term derived from symbolic dynamics [138]) if it can be presented by a definite graph. Thus the RLL constraint is finite type. There is also a useful intrinsic characterization of finite type constraints: there is an integer such that, for any symbol and any word of length at least , we have if and only if where is the suffix of of ....

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D. Lind and B. Marcus, Symbolic Dynamics and Coding. Cambridge, U.K.: Cambridge Univ. Press, 1995.

....(X, T ) must be exhibited or infinitely many congruences and infinitely many inequalities must be checked. It is therefore useful to have a small stock of well known examples of pairs (X, T ) for which f = f(T ) is known. These are standard: all the material below may be found in Lind and Marcus [8] (for the subshifts of finite type) and Chothi, Everest and Ward [4] for the group automorphisms) 7 Example 1.5 Given a matrix A # M k (Z ) there is an associated subshift of finite type (XA , TA ) which has exactly trace(A n ) points of period n. Two simple cases are worth ....

D. Lind and B. Marcus, Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.

.... the shift in this manner, thus yielding a surjection from the fundamental group of this space onto the group of automorphisms [2] For background on complex dynamics, we suggest the Proceedings from several previous AMS Short Courses [4] 5] For background on the relevant symbolic dynamics, see [8]. ....

Lind, D. and Marcus, B. Symbolic Dynamics and Coding. Cambridge Univ. Press, 1995.

....de Chile for nancial support and warm hospitality in his December 1998 visit to Santiago, which made this collaboration possible. 2. Symbolic background In this section we recall some elementary facts about symbolic dynamics. For a thorough introduction to the symbolic dynamics, see [K2] or [LM]. For a positive integer J , let J be a set of cardinality J ; our default choice will be f0; 1; J 1g. Let J denote the space Q n2Z J . We view a point x in J as a doubly in nite sequence of symbols from J , so x = x 1 x 0 x 1 : The space J is compact metrizable; one ....

....D. The images group Im(S) de ned in [BFF] is the quotient ZCO(X) H(S) To an n n integral matrix A, associate the direct limit group G(A) lim A Z n : The group G(A) can be presented concretely as a subgroup of a nite dimensional vector space (see pp. 14 15 of [BMT] and Sec. 7. 5 of [LM]) For a onesided SFT SA , we have Im(SA ) G(A) BFF] Thm. 4.5) For S = SN , let be the uniform measure (the measure of maximal entropy) i.e. if x[0; k] denotes fy 2 XN : y i = x i ; 0 i kg, then (x[0; k] N (k 1) Then there is an isomorphism Im(SN ) Z[1=N ] given by [ n i ....

D. Lind and B. Marcus, Symbolic Dynamics and Coding, Cambridge University, 1995.

....which do not consist of a periodic orbit; among them, 0 entropy as well as positive entropy systems with various properties. Other examples are less abstract: one sided Sturmian and Toeplitz systems are minimal subshifts, none of which is reduced to one periodic orbit (as a general reference see [LMa]) None of the above mentioned examples of noninvertible minimal maps is on a manifold. On the interval there is no minimal map at all and it is well known that the circle admits a minimal homeomorphism but does not admit any noninvertible minimal map (see [AK] In Section 3 we prove that in the ....

D. Lind and B. Marcus, Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, UK, 1995.

....the books by G. Edgar [E1] K. Falconer [F2] or P. Mattila [M] A good reference for dynamical systems theory is the book by C. Robinson [R] The code space and the shift map on it form the beginning of the theory of symbolic dynamics. See the books by B. Kitchens [K] and by D. Lind and B. Marcus [LM] for more on this subject. Let f1; Ng be the rst N integers. The code space will be denoted by = Q1 i=1 f1; Ng. The (full one sided) shift map is the map : de ned by ( i 1 ; i 2 ; i 3 ; i 2 ; i 3 ; ITERATED FUNCTION SYSTEMS AND THE CODE SPACE 3 ....

D. Lind and B. Marcus, Symbolic Dynamics and Coding, Cambridge, 1995.

....in that it can be de ned by a nite list of forbidden substrings, namely fbbg. Thus the language can be described in a purely local way; equivalently, we could list the allowed blocks faa; ab; bag of length 2. Sets of in nite sequences de ned in this way are called subshifts of nite type (e.g. [32]) Clearly any nite complement language is regular, but the reverse is not the case: for instance, a ba c) is regular but not f.c. since an in nite set of substrings ba b and ca c would have to be excluded. Whether the last non a was a b or a c is a hidden state , obscured ....

....analogy with one dimensional languages, we will usually construct languages of nite blocks, L [ m;n m;n ; however, we are also interested in sets of in nite con gurations, L1 . If these are closed and translationally invariant, they are called subshifts as in the one dimensional case (e.g. [32]) To translate back and forth between nite and in nite blocks, we introduce extension and restriction operators E and R. If L and L1 are languages of nite and in nite con gurations respectively, let E1 (L) fB 2 j 8b B : b 2 Lg R(L1 ) fb 2 [ m;n m;n j 9B 2 L1 : b Bg where by b ....

D. Lind and B. Marcus, Symbolic Dynamics and Coding. Cambridge University Press, 1995.

.... properties of the orbits can also be studied from the points of view of topology or measure theory [Luz93a] or representability through automata [BM86, Bla89, BF92] The case when A is a matrix is covered by the study of higher dimensional Bernoulli shifts, and more generally dynamical systems [Tak83, Sin89, LM95, Man87, dV93]. In this paper, we define the exact role played by m (the bound on the input variations) in the stability of (1) What we mean by stability will be precised below. Indeed, we look for the smallest m that guarantees stability for a scalar system (the matrix A is a scalar denoted a from now on) ....

D. Lind and B. Marcus. Symbolic Dynamics and Coding. Cambridge University Press, 1995.

....the number of cells in the grid grows exponentially with the number of dimensions 4 . Though the approximate nature of this representation does abstract away much detailed information about the dynamics, it preserves many of its important invariant properties; see, e.g. Hao[12] or Lind Marcus[20] for more details. This point is crucial to the phase portrait analysis methods used here; it means that conclusions drawn from the discretized trajectory are also true of the real trajectory that is, a repeating sequence of cells in the former, as in Figure 3, implies that the true dynamics ....

D. Lind and B. Marcus. Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.

....w n N Gamma1 ) for every n 2 Z. Then v n ; w n 2 P and v n = p( n ) w n ; h n ) 2 S( n ) for every 2 Gamma1 (fvg) and n 2 Z, so that v = 2 ( Omega ) This proves (4.11) Completion of the proof. According to (4. 8) the shift covariant surjective map is right resolving (cf. [11]) and hence j Gamma1 (v)j 2 P(A ThetaH) for every v 2 W . In particular, the restriction of to Omega is a continuous, boundedto one, shift covariant map of the SFT Omega onto W , and W is sofic. The remaining assertions follow from Proposition 4.1. ALGEBRAIC ....

D. Lind and B. Marcus, Symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995.

....An important property of closing factor maps is that they are always finite to one. An endomorphism of an irreducible SFT is surjective if and only if it is finite to one and consequently every closing endomorphism is surjective. For a thorough introduction to these topics, see [K2] or [LM]. 3. Algebraic maps In this section we consider factor maps which have an algebraic structure. This is the situation when the subshifts of finite type are also compact topological groups, the shift is a group automorphism and the factor map is a group homomorphism. An SFT which is also a ....

....(2) that it is an open map. So, a right resolving factor map between two one sided SFT s is a local homeomorphism. On the other hand, when is a finite to one factor map between irreducible SFT s, it is a consequence of the Perron Frobenius theorem that conditions (1) and (2) are equivalent. See [LM] Prop. 8.2.2 for (2) 1) the converse is similar. There is of course a similar definition of left resolving. The resolving maps have played a central role in the coding theory of symbolic dynamics ( K2] LM] A right resolving map is clearly right closing and modulo a recoding the converse is ....

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D. Lind and B. Marcus, Symbolic Dynamics and Coding, Cambridge University

....the full shift. The language of a shift X is the set B(X) of all subwords of sequences in X. Two shift spaces X and Y are conjugate if there is a one to one onto morphism OE : X Y which commutes with the shift transformation, OE ffi oe X = oe Y ffi OE. For more details, see Lind and Marcus [17]. To each automaton A = S A ; Delta A ; FA ) we associate three shift spaces (of bi infinite sequences) 1) the automaton shift, SA = f(q i ; a i ; x i ) i2Z j q i 2 SA ; a i 2 Sigma; x i 2 O; Delta A (q i ; a i ) q i 1 ; x i = FA (q i )g; 2) the label output shift, S Sigma;O A = f(a ....

....graphs (a property which is defined in the same way as for automata) This property corresponds to irreducibility of the sofic shifts. Recall that a shift X having the property that for every words u; v 2 B(X) there is a word w 2 B(X) such that uwv 2 B(X) is called irreducible. By Lind and Marcus [17], XG is irreducible iff G is strongly connected. We will now show that the label output shifts are actually sofic shifts. The idea is to transform the underlying automaton A into a labeled graph GA (over the alphabet Sigma Theta O) without affecting the represented shift. Example 5.1 Consider ....

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Lind, D., and Marcus, B. Symbolic Dynamics and Coding. Cambdridge University Press, Cambdridge, 1995.

....that it can be defined by a finite list of forbidden substrings, namely fbbg. Thus the language can be described in a purely local way; equivalently, we could list the allowed blocks faa; ab; bag of length 2. Sets of infinite sequences defined in this way are called subshifts of finite type (e.g. [27]) Clearly any finite complement language is regular, but the reverse is not the case: for instance, a ba c) is regular but not f.c. since an infinite set of substrings ba b and ca c would have to be excluded. Whether the last non a was a b or a c is a hidden state , obscured ....

....languages, we will usually construct languages of finite blocks, L ae [ m;n Sigma m;n ; however, we are also interested in sets of infinite configurations, L1 ae Sigma. If these are closed and translationally invariant, they are called subshifts as in the one dimensional case (e.g. [27]) To translate back and forth between finite and infinite blocks, we introduce extension and restriction operators E and R. If L and L1 are languages of finite and infinite configurations respectively, let E1 (L) fB 2 Sigma j 8b ae B : b 2 Lg R(L1 ) fb 2 [ m;n Sigma m;n j 9B 2 L1 : b ae ....

D. Lind and B. Marcus, Symbolic Dynamics and Coding. Cambridge University Press, 1995.

....de Chile for financial support and warm hospitality in his December 1998 visit to Santiago, which made this collaboration possible. 2. Symbolic background In this section we recall some elementary facts about symbolic dynamics. For a thorough introduction to the symbolic dynamics, see [K2] or [LM]. For a positive integer J , let J be a set of cardinality J ; our default choice will be f0; 1; J Gamma 1g. Let Sigma J denote the space Q n2Z J . We view a point x in Sigma J as a doubly infinite sequence of symbols from J , so x = x Gamma1 x 0 x 1 : The space Sigma J ....

....group Im(S) defined in [BFF] is the quotient ZCO(X) H(S) To an n Theta n integral matrix A, associate the direct limit group G(A) lim Gamma A Z n : The group G(A) can be presented concretely as a subgroup of a finite dimensional vector space (see pp. 14 15 of [BMT] and Sec. 7. 5 of [LM]) For a onesided SFT SA , we have Im(SA ) G(A) BFF] Thm. 4.5) For S = SN , let be the uniform measure (the measure of maximal entropy) i.e. if x[0; k] denotes fy 2 XN : y i = x i ; 0 i kg, then (x[0; k] N Gamma(k 1) Then there is an isomorphism Im(SN ) Z[1=N ] given by ....

D. Lind and B. Marcus, Symbolic Dynamics and Coding, Cambridge University, 1995.

....; h(z) n 0 : z 2 I0 ; i:e: z1 a) 1 : z 2 I1 ; i:e: z1 a) 6) depicted in Fig.6, then a unique binary sequence x = x(0) x(1) x(t) x(t 1) corresponds to each signal z. The symbol x(t) informs in which part (I0 or I1) the state z(t) is. This is called symbolic dynamic[4]. For the systems we have the facts: System Map (z1,z2, z2 1 0.5 0 0.5 1 1 0.5 0 0.5 1 1 0.5 0 0.5 1 z1 z2 f2(z1,z2) Figure 5. Graph of z2(t 1) 2 (z1(t) z2(t) 1 I I 0 1 1 1 a h(z1) z1 0 Figure 6. Graphs of the tent map and the encoder map Probability Density: Because each ....

....binary sequence s and an initial condition z(0) 2 Z in such a way that the symbolic dynamic x = x(0) x(1) of the chaotic signal z = z(0) z(1) is equal to the given sequence: s = x. This is due to the fact that the two interval partition Pi = fI0 ; I1g is a markov partition [4]. For instance a random sequence generated from a coin experiment corresponds to a chaotic signal z. Also a sequence s with a periodic pattern of length T , i.e. s(t T ) s(t) corresponds to an unstable periodic orbit of length T of the system. z (t) D D x(t L 1) x(t L 2) D x(t 1) x(t) D ....

Lind, D.; Marcus, B.: Symbolic Dynamics and Coding, Cambridge Univ.Press, 1995.

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D. Lind, B. Marcus, Symbolic Dynamics and Coding, Cambridge University Press, 1995.

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D. Lind, B. Marcus, Symbolic Dynamic and Coding, Cambridge University Press, Cambridge, 1995.

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D. Lind & B. Marcus, Symbolic Dynamics and Coding, Cambridge University Press, (1995).

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D. Lind and B. Marcus, Symbolic Dynamics and Coding. Cambridge, England: Cambridge University Press, 1995.

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D. Lind and B. Marcus, Symbolic Dynamics and Coding. Cambridge, England: Cambridge University Press, 1995.

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