For other systems, the diffraction spectrum can be a proper subset of the dynamical spectrum, as was pointed out for the Thue-Morse sequence (with singular continuous diffraction) by van Enter and Miȩkisz (J. dynamical spectrum (the opposite implication always being true). It is known that, under some mild assumptions, a pure point diffraction spectrum implies a pure point.
In general, the diffraction spectrum of ω, which is the carrier of the diffraction measure, lives on a subset of the dynamical spectrum. On a translation bounded measure ω leads to an interesting class of dynamical systems, with a rather rich spectral theory. Of the spectrum and tends to $1$ as $\lambda$ tends to $0$. The density of states measure is strictly smaller than the Hausdorff dimension Measure for an operator in this family is exact dimensional. Theorem are open and furthermore open linearly with respect to $\lambda$.Īdditionally, we show that, in the small coupling regime, the density of states Moreover, weĪlso show that at small coupling constant, all gaps allowed by the gap labeling This set tends to $1$ as coupling constant $\lambda$ tends to $0$. It is known that operators from this family have spectra which areĬantor sets of.
Sequences whose rotation angle has an eventually periodic continued fractionĮxpansion, a strictly larger class than primitive invertible substitution Potentials given by primitive invertible substitution sequences (or by Sturmian We study the spectral properties of discrete Schr\"odinger operators with In this article, we consider a twisted partial action $\unicodeG$. We alsoĭescribe an algorithm that computes the set of conjugacies between two such Group, and show that their one sided automorphism group is trivial. Prove the existence of an algorithm that computes their two sided automorphism Length substitutions which have metric discrete spectrum. We next focus on shifts generated by primitive constant The class of shifts defined by constant length primitive substitutions with aĬoincidence and with height one, we show that the two-sided automorphism group For a large class of Toeplitz shifts, including Group and the quotient automorphism group we give examples of shifts where The bound on theĬomplexity difference function gives a bound on the order of the automorphism One-sided shift is finite, and the quotient automorphism group of theĬorresponding two-sided shift "modulo the shift" is finite. Infinite minimal shift is bounded, then the the automorphism group of the We show that if the complexity difference function $p(n+1)-p(n)$ of an Indeed, the residually finite property of the group of automorphisms of the subshifts of finite type is a consequence of the existence of periodic points. This leads to the natural question about the relation between the algebraic properties of the group of automorphisms and the dynamics of the system. At the same time, they gave an example of a minimal Z-action on the Cantor set whose group of automorphisms contains Q, which implies that the automorphism group of a minimal action may be a non-residually finite group (recall that the Z-subshifts of finite type are not minimal). In, Boyle, Lind and Rudolph focused their attention on the group of automorphisms of subshifts of finite type, showing that these groups are always countable and residually finite. Recently, another algebraic invariant, the group of automorphisms of actions on the Cantor set, has caught the eye of several researchers working in the field.
Simon thomas minimal subshift simple amenable full#
On the other hand, the amenability of the topological full groups of minimal Z-actions together with their properties shown in by Matui make them the first known examples of infinite groups which are at the same time amenable, simple and finitely generated. Furthermore, the topological full group of such a system, which is known from Juschenko and Monod to be amenable, determines its flip-conjugacy class (see and for more details). For instance, Giordano, Putnam and Skau have shown in that the dimension group (see for an introduction about dimension groups) of a minimal Z-action on the Cantor set completely determines its strong orbit equivalence class. The algebraic invariants associated to the group actions on the Cantor set provide an interesting connection between the fields of dynamical systems and group theory.
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