Primitive Substitution Research Articles

On invertible substitutions with two fixed points

Let ϕ be a primitive substitution on a two-letter alphabet { a, b} having two fixed points ξ a and ξ b . We show that the substitution ϕ is invertible if and only if one has ξ a = abξ and ξ b = baξ. To cite this article: Z.-X. Wen et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 727–731.

Measure Zero Spectrum of a Class of Schrödinger Operators

We study the measure of the spectrum of a class of one-dimensional discrete Schrodinger operators Hv, ω with potential v(ω) generated by any primitive substitutions. It is well known that the spectrum of Hv, ω is singular continuous.(1) We will give a more exact result that the spectrum of Hv, ω is a set of Lebesgue measure zero, by removing two hypotheses (the semi-primitive of a certain induced substitution and the existence of square word) from a theorem due to Bovier and Ghez.(2)

Invertible substitutions and local isomorphisms

Let ϕ 1 and ϕ 2 be two primitive invertible substitutions over a two-letter alphabet. Let ξ ϕ 1 and ξ ϕ 2 be fixed points of ϕ 1 and ϕ 2, respectively. We show that ξ ϕ 1 and ξ ϕ 2 are locally isomorphic if and only if there exists a primitive invertible substitution ϕ 0 and two positive integers m and n such that M ϕ 1 = M ϕ 0 m and M ϕ 2 = M ϕ 0 n , where M ϕ is the substitutive matrix of the substitution ϕ. To cite this article: Z.-X. Wen et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 629–634.

A theorem of Cobham for non-primitive substitutions

In this article we generalize Cobham theorem to a large class of substitutions including non primitive and non constant length substitutions.

Directed Graphs and Substitutions

A substitution naturally determines a directed graph with an order- ing of the edges incident at each vertex. We describe a simple method by which any primitive substitution can be modified (without materially changing the bi- infinite fixed points of the substitution) so that points in the substitution minimal shift are in bijective correspondence with one-sided infinite paths on its associated directed graph. Using this correspondence, we show that primitive substitutive se- quences in the substitution minimal shift are precisely those sequences represented by eventually periodic paths. We use directed graphs to show that all measures of cylinders in a substitution minimal shift lie in a finite union of geometric se- quences, confirming a conjecture of Boshernitzan. Our methods also yield sufficient conditions for a geometric realization of a primitive substitution to be "almost injective." Every substitution has an associated prefix automaton, a directed graph whose vertices are the letters of the substitution's alphabet and whose edges are labeled with the strict prefixes of the images of the letters under the substitution. In this paper we consider the directed graph obtained by reversing the directed edges in the prefix automaton. This graph is more natural from the point of view of iterated function systems.

Pisot substitutions and Rauzy fractals

We prove that the dynamical system generated by a primitive unimodular substitution of the Pisot type ond letters satisfying a combinatorial condition which is easy to check, is measurably isomorphic to a domain exchange in R d 1 , and is a nite extension of a translation on the torus T d 1 .I n the course of the proof, we introduce some potentially useful notions: the linear maps associated to a substitution and their dual maps, and the -structure for a dynamical system with respect to a pair of partitions.

Substitution Hamiltonians with Bounded Trace Map Orbits

We investigate discrete one-dimensional Schrödinger operators with aperiodic potentials generated by primitive invertible substitutions on a two-letter alphabet. We prove that the spectrum coincides with the set of energies having a bounded trace map orbit and show that it is a Cantor set of zero Lebesgue measure. This result confirms a suggestion arising from a study of Roberts and complements results obtained by Bovier and Ghez. As an application we present a class of models exhibiting a purely singular continuous spectrum with probability one.

Palindrome complexity bounds for primitive substitution sequences

We consider one-sided infinite words generated via iteration by primitive substitutions on finite alphabets and provide bounds on the palindrome complexity function as well as uniform bounds on the frequencies of palindromes in such words. As an application of these bounds, we prove that the strongly palindromic sequences in a primitive substitution dynamical system form a set of measure zero.

Singular Continuous Spectrum for a Class of Substitution Hamiltonians II

We consider discrete one-dimensional Schrodinger operators with aperiodic potentials generated by primitive substitutions. Using the three-block version of Gordon's criterion, we establish purely singular continuous spectrum with probability one provided that the potentials have index greater than three. It is also shown that one cannot use this criterion to prove uniform results.

Descendants of Primitive Substitutions

Let s = (A, τ) be a primitive substitution. To each decomposition of the form τ (h) = uhv we associate a primitive substitution D[(h,u)](s) defined on the set of return words to h . The substitution D[(h,u)](s) is called a descendant of s and its associated dynamical system is the induced system (Xh, Th) on the cylinder determined by h . We show that \( {\cal D}(s) \) , the set of all descendants of s , is finite for each primitive substitution s . We consider this to be a symbolic counterpart to a theorem of Boshernitzan and Carroll which states that an interval exchange transformation defined over a quadratic field has only finitely many descendants. If s fixes a nonperiodic sequence, then \( {\cal D}(s) \) contains a recognizable substitution. Under certain conditions the set \( \Omega (s) = \bigcap_{s^\prime \in {\cal D}(s)}{\cal D}(s^\prime ) \) is nonempty.

A Generalization of Cobham's Theorem

If a nonperiodic sequence X is the image by a morphism of a fixed point of both a primitive substitution σ and a primitive substitution τ , then the dominant eigenvalues of the matrices of σ and τ are multiplicatively dependent. This is the way we propose to generalize Cobham's theorem.

Primitive substitutive numbers are closed under rational multiplication

Let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution. We show that the set M(r) is closed under multiplication by rational numbers, but not closed under addition.

On some properties of the decoding of admissible words for primitive substitutions

A number of properties of decoding for admissible sequences are studied, some of which were studied earlier, and some are new. Several conditions under which a substitution possesses these properties are obtained. Bibliography: 10 titles.

Singular continuous spectrum for palindromic Schrödinger operators

We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hullX of the potential is strictly ergodic, then the existence of just one potentialx inX for which the operator has no eigenvalues implies that there is a generic set inX for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such anx is that there is az∈X that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inX. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for allx∈X ifX derives from a primitive substitution. For potentials defined by circle maps,x n =1 J (θ0+nα), we show that the operator has purely singular continuous spectrum for a generic subset inX for all irrational α and every half-open intervalJ.

Some remarks on discrete aperiodic Schr�dinger operators

We consider Schrodinger operators onl2(ℤν) with deterministic aperiodic potential and Schrodinger operators on the l2-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl2(ℤν) fit into the formalism of ergodic random Schrodinger operators. Hence, their Lyapunov exponent, integrated density of states, and spectrum are almost-surely constant. We show that they are actually constant: the Lyapunov exponent for one-dimensional Schrodinger operators with potential defined by a primitive substitution, the integrated density of states, and the spectrum in arbitrary dimension if the system is strictly ergodic. We give examples of strictly ergodic Schrodinger operators that include several kinds of “almost-periodic” operators that have been studied in the literature. For Schrodinger operators on Penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular Penrose tiling under consideration.

Flexible architecture approach to knowledge-based analogue IC design

A new design methodology for knowledge-based analogue IC design is presented. Due to the nonlinear nature of the analogue design process, iterations are always necessary to achieve accurate and reliable designs. A prototype expert system design assistant and a conventional circuit simulator are used for the iterative design process. The expert system employed is capable of optimising circuit topologies, as well as circuit element geometries, to better satisfy the performance specifications. Circuit reconstruction is achieved through circuit primitive replacements and design equation substitutions. Circuit primitive and critical nodes are processed before layout generation to minimise undesired parasitic effects. Layout parasitics can be included optionally in the design optimisation step. Experimental results are shown, to demonstrate the effectiveness of the flexible architecture approach through iteration and the expert analogue design system.

Scientific Serial

American Journal of Mathernagics, vol. xxiii. No. 3.—Geometry on the cubic scroll of the second kind, by F. C. Ferry, is the conclusion (34 pp.) of a paper commenced in the last number.—Congruent reductions of bilinear forms, by T. J. I'A. Bromwich, contains an account and a slight extension of a method due to Kronecker (Gesamm. Werke, Bd. i. p. 349). This method was employed in the first place for the reduction of two quadratic forms. In the present paper it is applied to four cases of reductions, viz. (1) two symmetric forms (the same as Kronecker's case); (2) a symmetric and an alternate form (3) two alternate forms; and (4) two Hermite's forms. In cases (1)–(3) the substitutions are congruent, while in (4) they are conjugate imaginaries. Mr. Bromwich gives a list of the principal papers which deal with the problems he has considered in his article. On the imprimitive substitution groups of degree fifteen and the primitive substitution groups of degree eighteen, by E. Norton Martin, was presented, in abstract and in a slightly different form, at the summer meeting of the American Mathematical Society in 1899. Herein he has added two new groups to his original list, viz. the groups with five systems of imprimitivity simply isomorphic to the alternating and symmetric groups of degree five, and he mentions that Dr. Kuhn reported at the February (1900) meeting of the Society that he had carried the investigation further by adding twenty-eight to the seventy groups found by Mr. Martin. The list even now does not claim to be absolutely complete, since omissions are always possible. A somewhat long list of recent papers on the subject is appended to the article.—Removal of any two terms from a binary quantic by linear transformations, by Bessie G. Morrison, discusses these linear transformations and gives applications to the non-singular cubic, quartic, quintic and sextic.

On the Imprimitive Substitution Groups of Degree Fifteen and the Primitive Substitution Groups of Degree Eighteen

On the Imprimitive Substitution Groups of Degree Fifteen and the Primitive Substitution Groups of Degree Eighteen

On the Primitive Substitution Groups of Degree Fifteen

Proceedings of the London Mathematical SocietyVolume s1-28, Issue 1 p. 533-544 Articles On the Primitive Substitution Groups of Degree Fifteen G. A. Miller Ph.D., G. A. Miller Ph.D.Search for more papers by this author G. A. Miller Ph.D., G. A. Miller Ph.D.Search for more papers by this author First published: November 1896 https://doi.org/10.1112/plms/s1-28.1.533Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes1-28, Issue1November 1896Pages 533-544 RelatedInformation