Systemes de numeration et fonctions fractales relatifs aux substitutions

AbstractLet A be a finite alphabet, σ a substitution over A, (un)n ϵ N a fixed point for σ, and for each a ϵ A, ƒ(a) a real number. We establish, under some assumptions, an asymptotic formula concerning the sum Sƒ (N) = Σi ⩽ N ƒ(ui), N ϵ N. This result generalizes some previous results from Coquet or Brillhart, Erdös, and Morton. Moreover, relations with self-affine functions (in a sense which generalizes a definition from Kamae) are proved. The calculi leave over systems of representation of integers and real numbers

Semigroups and the self-similar structure of the flipped tribonacci substitution

AbstractUsing the numeration system associated to the substitution 1 → 12, 2 → 31, 3 → 1, we define a binary operation, which generates a semigroup on a subset of N. This semigroup represents the self-similar structure defined by an Iterated Function System (IFS) on the dynamical system associated to this substitution

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

Joint Spectral Radius, Dilation Equations, and Asymptotic Behavior of Radix-Rational Sequences

International audienceRadix-rational sequences are solutions of systems of recurrence equations based on the radix representation of the index. For each radix-rational sequence with complex values we provide an asymptotic expansion, essentially in the scale N^α log^l N. The precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence of first-order differences. The coefficients are Hölderian functions obtained through some dilation equations, which are usual in the domains of wavelets and refinement schemes. The proofs are ultimately based on elementary linear algebra.Les suites rationnelles dans une base de numération sont solutions de récurrences basées sur la représentation de l'indice dans la base de numération. Pour chacune de ces suites, à valeurs complexes, nous fournissons un développement asymptotique, essentiellement dans l'échelle N^α log^l N. La précision du développement dépend du rayon spectral de la représentation linéaire pour la suite des différences de la suite considérée. Les coefficients sont des fonctions höldériennes obtenues par des équations de dilatation, usuelles dans les théories des ondelettes et des schémas de raffinement. Les preuves sont basées sur l'algèbre linéaire élémentaire

The geometry of non-unit Pisot substitutions

Let $\sigma$ be a non-unit Pisot substitution and let $\alpha$ be the associated Pisot number. It is known that one can associate certain fractal tiles, so-called \emph{Rauzy fractals}, with $\sigma$. In our setting, these fractals are subsets of a certain open subring of the ad\`ele ring $\mathbb{A}_{\mathbb{Q}(\alpha)}$. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, define them in terms of the one-dimensional realization of $\sigma$ and its dual (in the spirit of Arnoux and Ito), and view them as the dual of multi-component model sets for particular cut and project schemes. We also define stepped surfaces suited for non-unit Pisot substitutions. We provide basic topological and geometric properties of Rauzy fractals associated with non-unit Pisot substitutions, prove some tiling results for them, and provide relations to subshifts defined in terms of the periodic points of $\sigma$, to adic transformations, and a domain exchange. We illustrate our results by examples on two and three letter substitutions.Comment: 29 page

Mean asymptotic behaviour of radix-rational sequences and dilation equations (Extended version)

The generating series of a radix-rational sequence is a rational formal power series from formal language theory viewed through a fixed radix numeration system. For each radix-rational sequence with complex values we provide an asymptotic expansion for the sequence of its Ces\`aro means. The precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence; the coefficients are obtained through some dilation equations. The proofs are based on elementary linear algebra

Almost everywhere balanced sequences of complexity 2n+12n+1

We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set $\{1,2\}^\mathbb{N}$ of directive sequences. For a given set $\mathcal{C}$ of two substitutions, we show that there exists a $\mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $\mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $\mu$-almost every $\mathcal{C}$-adic sequence is balanced, where $\mu$ is any shift-invariant ergodic Borel probability measure on $\{1,2\}^\mathbb{N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure $\mu$ is negative.Comment: 42 pages, 9 figures. Extended and augmented version of arXiv:1707.0274