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For a class of -interval exchange transformations, which we call the symmetric class, we define a new self-dual induction process in which the system is successively induced on a union of sub-intervals. This algorithm gives rise to an underlying graph structure which reflects the dynamical behavior of the system, through the Rokhlin towers of the induced maps. We apply it to build a wide assortment of explicit examples on four intervals having different dynamical properties: these include the first...
We consider the problem of embedding odometers in one-dimensional cellular automata. We show that (1) every odometer can be embedded in a gliders-with-reflecting-walls cellular automaton, which one depending on the odometer, and (2) an odometer can be embedded in a cellular automaton with local rule (i ∈ ℤ), where n depends on the odometer, if and only if it is “finitary.”
We show that an aperiodic minimal tiling space with only finitely many asymptotic composants embeds in a surface if and only if it is the suspension of a symbolic interval exchange transformation (possibly with reversals). We give two necessary conditions for an aperiodic primitive substitution tiling space to embed in a surface. In the case of substitutions on two symbols our classification is nearly complete. The results characterize the codimension one hyperbolic attractors of surface diffeomorphisms...
We associate with a word on a finite alphabet an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of . Then when we deduce, using the sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.
We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of w. Then when |A|=2 we deduce, using the Sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.
A (quadratic) Hubbard tree is an invariant tree connecting the critical orbit within the Julia set of a postcritically finite (quadratic) polynomial. It is easy to read off the kneading sequences from a quadratic Hubbard tree; the result in this paper handles the converse direction. Not every sequence on two symbols is realized as the kneading sequence of a real or complex quadratic polynomial. Milnor and Thurston classified all real-admissible sequences, and we give a classification of all complex-admissible...
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