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Richomme, G., Saari, K., Zamboni, L.Q.: Abelian complexity of minimal subshifts. Puzynina, S., Zamboni, L.Q.: Abelian returns in Sturmian words. We also obtained a quantitative upper bound for the measure of the spectrum. uniformly recurrent and generate the same minimal subshift, we call it the substi-. Abstract: We show that the measure of the spectrum of Schr\'odinger operator with potential defined by non-constant function over any minimal aperiodic finite subshift tends to zero, as the coupling constant tends to infinity. A minimal subshift (X, T) is linearly recurren t if there exists a constan t K so that f o r eac h clop en set U generated by a nite w ord u the return time to U, with resp ect to T, is b. Morse, M., Hedlund, G.A.: Symbolic dynamics II. two multiplicatively independent integers and A be a finite alphabet. Cambridge University Press, Cambridge (2002). Lothaire, M.: Algebraic Combinatorics on Words, vol. Encyclopedia of Mathematics and Its Applications, vol. By Lemma 2.2 we can assume that is a right-infinite subshift: if is bi-infinite we consider its right-infinite restriction +, if is left-infinite we simply consider its right-infinite mirror image. Then has bounded powers if only if sup d inf are Lipschitz equivalent. The aim of this section is to prove that every commutator subgroup of a Cantor minimal subshift is nitely generated, Theorem 0.0.3. Let be a minimal aperiodic subshift over a finite alphabet. Cambridge University Press, New York (1995) Finite generation of the commutator subgroup. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Kaboré, I., Tapsoba, T.: Combinatoire de mots récurrents decomplexité \(n+2\). Hejda, T., Steiner, W., Zamboni, L.Q.: What is the Abelianization of the Tribonacci shift? In: Workshop on Automatic Sequences, Liége (2015). For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.Two finite words u and v are called Abelian equivalent if each letter occurs equally many times in both u and v. Minimal systems are natural generalizations of periodic orbits, and they are analogues of ergodic measures in topological dynamics.They were defined by G. Examples include the “integer Sierpinski gasket and carpet” tilings. A key step is to establish recognizability of non-periodic tilings in our setting. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. We adopt this point of view and define subshifts of finite type via the Parry-Bowen-Lanford matrix, A. This matrix essentially dates from the earlier papers of Parry 9, 10 on intrinsic Markov chains. In this paper we continue the study of linearly recurrent (LR) subshifts initi-ated in DHS. Since then, generalisations and extensions of these features, namely \\alpha -repetitive, \\alpha -repulsive and \\alpha -finite ( \\alpha \\ge 1. We also give a constructive characterization of these subshifts. At the turn of this century Durand, and Lagarias and Pleasants established that key features of minimal subshifts (and their higher-dimensional analogues) to be studied are linearly repetitive, repulsive and power free. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. A (see 1) to each subshift of finite type. a linearly recurrent subshift (X,T) the set of its non-periodic subshift factors is nite up to isomorphism. By a minimal subshift we mean one which is minimal with respect to. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. 9.2 Theorem For irreducible subshifts, the profinite group G(X) is a conjugacy invariant. It is well known that in the primitive case, the dynamical system is uniquely ergodic. Abstract: We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems.