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A characterization of ω-limit sets for piecewise monotone maps of the interval

Andrew D. Barwell (2010)

Fundamenta Mathematicae

For a piecewise monotone map f on a compact interval I, we characterize the ω-limit sets that are bounded away from the post-critical points of f. If the pre-critical points of f are dense, for example when f is locally eventually onto, and Λ ⊂ I is closed, invariant and contains no post-critical point, then Λ is the ω-limit set of a point in I if and only if Λ is internally chain transitive in the sense of Hirsch, Smith and Zhao; the proof relies upon symbolic dynamics. By identifying points of...

A class of tridiagonal operators associated to some subshifts

Christian Hernández-Becerra, Benjamín A. Itzá-Ortiz (2016)

Open Mathematics

We consider a class of tridiagonal operators induced by not necessary pseudoergodic biinfinite sequences. Using only elementary techniques we prove that the numerical range of such operators is contained in the convex hull of the union of the numerical ranges of the operators corresponding to the constant biinfinite sequences; whilst the other inclusion is shown to hold when the constant sequences belong to the subshift generated by the given biinfinite sequence. Applying recent results by S. N....

A contribution to infinite disjoint covering systems

János Barát, Péter P. Varjú (2005)

Journal de Théorie des Nombres de Bordeaux

Let the collection of arithmetic sequences { d i n + b i : n ∈ ℤ } i ∈ I be a disjoint covering system of the integers. We prove that if d i = p k q l for some primes p , q and integers k , l ≥ 0 , then there is a j ≠ i such that d i | d j . We conjecture that the divisibility result holds for all moduli.A disjoint covering system is called saturated if the sum of the reciprocals of the moduli is equal to 1 . The above conjecture holds for saturated systems with d i such that the product of its prime factors is at most 1254 .

A generalization of the self-dual induction to every interval exchange transformation

Sébastien Ferenczi (2014)

Annales de l’institut Fourier

We generalize to all interval exchanges the induction algorithm defined by Ferenczi and Zamboni for a particular class. Each interval exchange corresponds to an infinite path in a graph whose vertices are certain unions of trees we call castle forests. We use it to describe those words obtained by coding trajectories and give an explicit representation of the system by Rokhlin towers. As an application, we build the first known example of a weakly mixing interval exchange outside the hyperelliptic...

A graph approach to computing nondeterminacy in substitutional dynamical systems

Toke M. Carlsen, Søren Eilers (2007)

RAIRO - Theoretical Informatics and Applications

We present an algorithm which for any aperiodic and primitive substitution outputs a finite representation of each special word in the shift space associated to that substitution, and determines when such representations are equivalent under orbit and shift tail equivalence. The algorithm has been implemented and applied in the study of certain new invariants for flow equivalence of substitutional dynamical systems.

A map maintaining the orbits of a given ℤ d -action

Bartosz Frej, Agata Kwaśnicka (2016)

Colloquium Mathematicae

Giordano et al. (2010) showed that every minimal free ℤ d -action of a Cantor space X is orbit equivalent to some ℤ-action. Trying to avoid the K-theory used there and modifying Forrest’s (2000) construction of a Bratteli diagram, we show how to define a (one-dimensional) continuous and injective map F on X∖one point such that for a residual subset of X the orbits of F are the same as the orbits of a given minimal free ℤ d -action.

A matrix formalism for conjugacies of higher-dimensional shifts of finite type

Michael Schraudner (2008)

Colloquium Mathematicae

We develop a natural matrix formalism for state splittings and amalgamations of higher-dimensional subshifts of finite type which extends the common notion of strong shift equivalence of ℤ⁺-matrices. Using the decomposition theorem every topological conjugacy between two ℤ d -shifts of finite type can thus be factorized into a finite chain of matrix transformations acting on the transition matrices of the two subshifts. Our results may be used algorithmically in computer explorations on topological...

A new algebraic invariant for weak equivalence of sofic subshifts

Laura Chaubard, Alfredo Costa (2008)

RAIRO - Theoretical Informatics and Applications

It is studied how taking the inverse image by a sliding block code affects the syntactic semigroup of a sofic subshift. The main tool are ζ-semigroups, considered as recognition structures for sofic subshifts. A new algebraic invariant is obtained for weak equivalence of sofic subshifts, by determining which classes of sofic subshifts naturally defined by pseudovarieties of finite semigroups are closed under weak equivalence. Among such classes are the classes of almost finite type subshifts...

A note on a conjecture of Duval and sturmian words

Filippo Mignosi, Luca Q. Zamboni (2002)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We prove a long standing conjecture of Duval in the special case of sturmian words.

A remark on the topological entropies of covers and partitions

Pierre-Paul Romagnoli (2007)

Studia Mathematica

We study if the combinatorial entropy of a finite cover can be computed using finite partitions finer than the cover. This relates to an unsolved question in [R] for open covers. We explicitly compute the topological entropy of a fixed clopen cover showing that it is smaller than the infimum of the topological entropy of all finer clopen partitions.

A study on global stabilization of periodic orbits in discrete-time chaotic systems by using symbolic dynamics

Masayasu Suzuki, Noboru Sakamoto (2015)

Kybernetika

In this report, a control method for the stabilization of periodic orbits for a class of one- and two-dimensional discrete-time systems that are topologically conjugate to symbolic dynamical systems is proposed and applied to a population model in an ecosystem and the Smale horseshoe map. A periodic orbit is assigned as a target by giving a sequence in which symbols have periodicity. As a consequence, it is shown that any periodic orbits can be globally stabilized by using arbitrarily small control...

A survey on transitivity in discrete time dynamical systems. application to symbolic systems and related languages

Gianpiero Cattaneo, Alberto Dennunzio, Fabio Farina (2006)

RAIRO - Theoretical Informatics and Applications

The main goal of this paper is the investigation of a relevant property which appears in the various definition of deterministic topological chaos for discrete time dynamical system: transitivity. Starting from the standard Devaney's notion of topological chaos based on regularity, transitivity, and sensitivity to the initial conditions, the critique formulated by Knudsen is taken into account in order to exclude periodic chaos from this definition. Transitivity (or some stronger versions of it)...

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