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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...
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....
Let the collection of arithmetic sequences be a disjoint covering system of the integers. We prove that if for some primes and integers , then there is a such that . 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 . The above conjecture holds for saturated systems with such that the product of its prime factors is at most .
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...
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.
Giordano et al. (2010) showed that every minimal free -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 -action.
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 -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 natural occcurrence of shift equivalence in a purely algebraic setting is exhibited.
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...
We prove a long standing conjecture of Duval in the special case of sturmian words.
We prove a long standing conjecture of Duval in the special case
of
Sturmian words.
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.
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...
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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