82 research outputs found
Invariant measures, matching and the frequency of 0 for signed binary expansions
We introduce a parametrised family of maps ,
called symmetric doubling maps, defined on by ,
where . Each map generates binary expansions with
digits , 0 and 1. We study the frequency of the digit 0 in typical
expansions as a function of the parameter . The transformations
have a natural ergodic invariant measure that is absolutely
continuous with respect to Lebesgue measure. The frequency of the digit 0 is
related to the measure by the Ergodic Theorem.
We show that the density of is piecewise smooth except for a set of
parameters of zero Lebesgue measure and full Hausdorff dimension and give a
full description of the structure of the maximal parameter intervals on which
the density is piecewise smooth. We give an explicit formula for the frequency
of the digit 0 in typical signed binary expansions on each of these parameter
intervals and show that this frequency depends continuously on the parameter
. Moreover, it takes the value only on the interval and it is strictly less than on the remainder
of the parameter space.Comment: 30 pages, 4 figure
Some properties of Ising automata
AbstractIn this work, we shall present some arithmetical and topological properties of Ising automata. More precisely, we shall study many different notions, such as faithful and strictly faithful automata, factor and product automata, irreducible and weakly irreducible automata, prime automata, homogeneous automata, minimal automata, invertible automata, etc., and discuss their related properties. We shall also define and study three different topologies over the set of all minimal automata, and discuss the topological closure property of automatic sequences. As application, we shall use the obtained results to give a somewhat detailed analysis of Ising automata
Rational series and asymptotic expansion for linear homogeneous divide-and-conquer recurrences
Among all sequences that satisfy a divide-and-conquer recurrence, the
sequences that are rational with respect to a numeration system are certainly
the most immediate and most essential. Nevertheless, until recently they have
not been studied from the asymptotic standpoint. We show how a mechanical
process permits to compute their asymptotic expansion. It is based on linear
algebra, with Jordan normal form, joint spectral radius, and dilation
equations. The method is compared with the analytic number theory approach,
based on Dirichlet series and residues, and new ways to compute the Fourier
series of the periodic functions involved in the expansion are developed. The
article comes with an extended bibliography
Spectral Properties of Schr\"odinger Operators Arising in the Study of Quasicrystals
We survey results that have been obtained for self-adjoint operators, and
especially Schr\"odinger operators, associated with mathematical models of
quasicrystals. After presenting general results that hold in arbitrary
dimensions, we focus our attention on the one-dimensional case, and in
particular on several key examples. The most prominent of these is the
Fibonacci Hamiltonian, for which much is known by now and to which an entire
section is devoted here. Other examples that are discussed in detail are given
by the more general class of Schr\"odinger operators with Sturmian potentials.
We put some emphasis on the methods that have been introduced quite recently in
the study of these operators, many of them coming from hyperbolic dynamics. We
conclude with a multitude of numerical calculations that illustrate the
validity of the known rigorous results and suggest conjectures for further
exploration.Comment: 56 page
Symmetry and Asymmetry in Quasicrystals or Amorphous Materials
About forty years after its discovery, it is still common to read in the literature that quasicrystals (QCs) occupy an intermediate position between amorphous materials and periodic crystals. However, QCs exhibit high-quality diffraction patterns containing a collection of discrete Bragg reflections at variance with amorphous phases. Accordingly, these materials must be properly regarded as long-range ordered materials with a symmetry incompatible with translation invariance. This misleading conceptual status can probably arise from the use of notions borrowed from the amorphous solids framework (such us tunneling states, weak interference effects, variable range hopping, or spin glass) in order to explain certain physical properties observed in QCs. On the other hand, the absence of a general, full-fledged theory of quasiperiodic systems certainly makes it difficult to clearly distinguish the features related to short-range order atomic arrangements from those stemming from long-range order correlations. The contributions collected in this book aim at gaining a deeper understanding on the relationship between the underlying structural order and the resulting physical properties in several illustrative aperiodic systems, including the border line between QCs and related complex metallic alloys, hierarchical superlattices, electrical transmission lines, nucleic acid sequences, photonic quasicrystals, and optical devices based on aperiodic order designs
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