Symbolic Dynamical Systems Research Articles

WHAT MAPS CAN ADMIT TWO-SIDED SYMBOLIC DYNAMICAL SYSTEMS?

A continuous map f from a metric space X to itself is said to contain a two-sided symbolic dynamical system if there exists an invariant set X0 of f such that the subsystem f|X0 is topologically conjugate to the shift map on a two-sided sequence space of some symbols. In this paper we show that, for any given integer n ≥ 2, there exists a Lipschitz continuous interval map which contains a two-sided symbolic dynamical system of n symbols. Furthermore, we investigate the effect of differentiability and monotonicity assumptions, and prove that neither piecewise monotonic nor piecewise continuously differentiable graph map can contain a two-sided symbolic dynamical system.

Embedding of nonlinear dynamical systems with compact and totally disconnected domains into the product symbolic dynamical systems

It is known that any ɛ -expansive dynamical system with compact and totally disconnected domain can be embedded into a certain symbolic dynamical system. Actually, there exists a dynamical system with a compact and totally disconnected domain that cannot be topologically embedded into any symbolic dynamical system. Exactly speaking, ɛ -expansiveness plays an important role in the embedding problem. In this paper, the embedding theorem of dynamical systems which are not ɛ -expansive is discussed, and a relation between this result and the topological entropy is given.

Connectedness of number theoretical tilings

Let T=T(A,D) be a self-affine tile in \reals^n defined by an integral expanding matrix A and a digit set D. In connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \0,1,..., |det(A)|-1\. It is shown that in \reals^3 and \reals^4, for any integral expanding matrix A, T(A,D) is connected. We also study the connectedness of Pisot dual tilings which play an important role in the study of β -expansion, substitution and symbolic dynamical system. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, a complete classification of the β -expansion of 1 for quartic Pisot units is given.

On the expressiveness and decidability of o-minimal hybrid systems

This paper is driven by a general motto: bisimulate a hybrid system by a finite symbolic dynamical system. In the case of o-minimal hybrid systems, the continuous and discrete components can be decoupled, and hence, the problem reduces in building a finite symbolic dynamical system for the continuous dynamics of each location. We show that this can be done for a quite general class of hybrid systems defined on o-minimal structures. In particular, we recover the main result of a paper by G. Lafferriere, G.J. Pappas, and S. Sastry, on o-minimal hybrid systems. We also provide an analysis and extension of results on decidability and complexity of problems and constructions related to o-minimal hybrid systems.

On almost automorphic dynamics in symbolic lattices

We study the existence, structure and topological entropy of almost automorphic arrays in symbolic lattice dynamical systems. In particular, we show that almost automorphic arrays with arbitrarily large entropy are typical in symbolic lattice dynamical systems. Applications to pattern formation and spatial chaos in infinite-dimensional lattice systems are considered and the construction of chaotic almost automorphic signals is discussed.

Strong shift equivalence of symbolic dynamical systems and Morita equivalence of C *-algebras

Symbolic matrix systems are generalizations of finite symbolic matrices for sofic systems to subshifts. We prove that if two symbolic matrix systems are strong shift equivalent, then the gauge actions of the associated C*-algebras are stably outer conjugate. The proof given here is based on the construction of an imprimitivity bimodule from a bipartite -graph system, so that an equivariant version of the Brown–Green–Rieffel Theorem proved by Combes is used, together with its proof. As a corollary, if two subshifts are topologically conjugate, then the gauge actions of the associated C*-algebras are stably outer conjugate.

Symbolic stochastic dynamical systems viewed as binary N-step Markov chains.

A theory of systems with long-range correlations based on the consideration of binary N-step Markov chains is developed. In the model, the conditional probability that the ith symbol in the chain equals zero (or unity) is a linear function of the number of unities among the preceding N symbols. The correlation and distribution functions as well as the variance of the number of symbols in the words of arbitrary length L are obtained analytically and numerically. A self-similarity of the studied stochastic process is revealed and the similarity group transformation of the chain parameters is presented. The diffusion Fokker-Planck equation governing the distribution function of the L words is explored. If the persistent correlations are not extremely strong, the distribution function is shown to be the Gaussian with the variance being nonlinearly dependent on L. The applicability of the developed theory to the coarse-grained written and DNA texts is discussed.

A twisted tensor product on symbolic dynamical systems and the Ashley's problem

We define the notion of fiber bundle via a twisted tensor product on the transition matrices. We define the notion of topological conjugacy and shift equivalence in this bundle context and show that topological conjugacy implies shift equivalence. We show that the $\Sigma_A$ fits into our fiber bundle context. We introduce another system $\Sigma_W$, topologically conjugate to the full $2-$shift, which has the same base space and fiber as the Ashley system, but is constructed with a different twisting. We show that $\Sigma_A$ and $\Sigma_W$ are shift equivalent but not bundle isomorphic.

Generic points in the cartesian powers of the Morse dynamical system

The symbolic dynamical system associated with the Morse sequence is strictly ergodic. We describe some topological and metrical properties of the Cartesian powers of this system, and some of its other self-joinings. Among other things, we show that non generic points appear in the fourth power of the system, but not in lower powers. We exhibit various examples and counterexamples related to the property of weak disjointness of measure preserving dynamical systems.

Generalized Fermat, double Fermat and Newton sequences

In this paper, we discuss the relationship among the generalized Fermat, double Fermat, and Newton sequences. In particular, we show that every double Fermat sequence is a generalized Fermat sequence, and the set of generalized Fermat sequences, as well as the set of double Fermat sequences, is closed under term-by-term multiplication. We also prove that every Newton sequence is a generalized Fermat sequence and vice versa. Finally, we show that double Fermat sequences are Newton sequences generated by certain sequences of integers. An approach of symbolic dynamical systems is used to obtain congruence identities.

Minimal actions of the group $ {\Bbb S(Z)} $ of permutations of the integers

Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact non-compact group this is a nonmetrizable system with a very rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. One such group is the topological group $ \Bbb S $ of all the permutations of the integers $ \Bbb Z $ , with the topology of pointwise convergence. In this paper we show that (M( $ \Bbb S $ ), $ \Bbb S $ ) is a symbolic dynamical system (hence in particular M( $ \Bbb S $ ) is a Cantor set), and we give a full description of all its symbolic factors.

Multifractal properties of Hao's geometric representations of DNA sequences

Hao proposed a graphic representation of subsequence structure in DNA sequences and computed fractal dimensions of such representations for factorizable languages. In this study, we extend Hao's work in several directions: (1) We generalize Hao's scheme to accommodate sequences over an arbitrary finite number of symbols. (2) We establish a direct correspondence between the statistical characterization of symbolic sequences via Rényi entropy spectra and the multifractal characteristics (Rényi generalized dimensions) of the sequences’ spatial representations. (3) We show that for general symbolic dynamical systems, the multifractal f H -spectra in the sequence space endowed with commonly used metrics, coincide with the f H -spectra on Hao's sequence representations. (4) So far the connection between the Hao's scheme and another well-known subsequence visualization scheme—Jeffrey's chaos game representation (CGR)—has been characterized only in very vague terms. We show that the fractal dimension results for Hao's visualization frames directly translate to Jeffrey's CGR scheme.

Bowen–Franks groups as an invariant for flow equivalence of subshifts

It will be proved that some topological conjugacy invariants, called K-groups and Bowen–Franks groups, for subshifts are invariants for flow equivalence by an argument of symbolic dynamical systems. Some examples of non-sofic subshifts that are not flow equivalent will be presented.

Determining presentations of sofic shifts

A sofic shift is a symbolic dynamical system which consists of all sequences of labels occurring in bi-infinite paths in a finite labelled directed graph. The graph defines a presentation of the sofic shift. Determining presentations of a sofic shift S are a generalization of resolving (or deterministic) presentations. The collection of minimal determining presentations is a conjugacy invariant of~S. We give a procedure for constructing all minimal determining presentations of an irreducible sofic shift, which is a generalization of the construction of the Fischer covers of~S.

Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension

For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a non-trivial mathematical application of this theory.

Embedding of expansive dynamical systems into symbolic dynamical systems

Let ( X, d) be a compact and totally disconnected metric space, and T an expansive continuous mapping on X with values in X. Then, it is shown that the expansive dynamical system ( X, d, T) can be embedded into a certain symbolic dynamical system. Moreover, compactification of the family consisting of all nonexpansive mappings on a compact metric space is given.

Geodesic laminations as geometric realizations of Pisot substitutions

We construct a geodesic lamination on the hyperbolic disk and a dynamical system defined on this lamination. We prove that this dynamical system is a geometrical realization of the symbolic dynamical system that arises from the following Pisot substitution: $1\rightarrow 12, \dotsc, (n-1) \rightarrow 1n, n\rightarrow 1$.

Forbidden Words in Symbolic Dynamics

We introduce an equivalence relation≃between functions from N to N. By describing a symbolic dynamical system in terms of forbidden words, we prove that the≃-equivalence class of the function that counts the minimal forbidden words of a system is a topological invariant of the system. We show that the new invariant is independent from previous ones, but it is not characteristic. In the case of sofic systems, we prove that the≃-equivalence of the corresponding functions is a decidable question. As a more special application, we show, by using the new invariant, that two systems associated to Sturmian words having “different slope” are not conjugate.

Coloring link diagrams with a continuous palette

The well-known technique of n-coloring a diagram of an oriented link l is generalized using elements of the circle T for colors. For any positive integer r, the more general notion of a ( T , r )-coloring is defined by labeling the arcs of a diagram D with elements of the torus T r−1 . The set of ( T , r )-colorings of D is an abelian group, and its quotient by the connected component of the identity is isomorphic to the torsion subgroup of H 1(M r(l) ; Z). Here M r(l) denotes the r-fold cyclic cover of S 3 branched over the link l. Results about braid entropy are obtained using techniques of symbolic dynamical systems.

Stabilized $C^*$-algebras constructed from symbolic dynamical systems

We construct stabilized $C^*$-algebras from subshifts by using the dynamical property of the symbolic dynamical systems. We prove that the construction is dynamical and the $C^*$-algebras are isomorphic to the tensor product $C^*$-algebras between the algebra of all compact operators on a separable Hilbert space and the $C^*$-algebras constructed from creation operators on sub-Fock spaces associated with the subshifts. We also prove that the gauge actions on the stabilized $C^*$-algebras are invariant for topological conjugacy as two-sided subshifts under some conditions. Hence, if two subshifts are topologically conjugate as two-sided subshifts, the associated stabilized $C^*$-algebras are isomorphic so that their K-groups are isomorphic.