Automatic sequences based on Parry or Bertrand numeration systems

We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand numeration systems. These automatic sequences can be viewed as generalizations of the more typical k-automatic sequences and Pisot-automatic sequences. We show that, like k-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry numeration system is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for k-automatic sequences and Pisot-automatic sequences, so our result shows that these properties are lost when generalizing to Parry numeration systems and beyond. Moreover, we show that a multidimensional sequence is U -automatic with respect to a positional numeration system U with regular language of numeration if and only if its U -kernel is finite

Automatic sequences based on Parry or Bertrand numeration systems

We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand numeration systems. These automatic sequences can be viewed as generalizations of the more typical $k$-automatic sequences and Pisot-automatic sequences. We show that, like $k$-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry numeration system is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for $k$-automatic sequences and Pisot-automatic sequences, so our result shows that these properties are lost when generalizing to Parry numeration systems and beyond. Moreover, we show that a multidimensional sequence is $U$-automatic with respect to a positional numeration system $U$ with regular language of numeration if and only if its $U$-kernel is finite.</p

Positional Numeration Systems: Ultimate Periodicity, Complexity and Automatic Sequences

This dissertation thesis is made up of three distinct parts, connected especially by complexity notion, factorial complexity as well as state complexity. We study positional numeration systems and recognizable sets through decision problems and automatic sequences. The first part is devoted to the following problem: given a numeration system U and a finite automaton accepting U-representations of a set X ⊆ N, can we decide whether the set X is ultimately periodic (i.e. a finite union of arithmetic progressions)? We prove that this problem is decidable for a large class of numeration systems based on linear recurrent sequences. Thanks to the given automaton, we bound the possible periods of X via an arithmetical study of the linear recurrent sequence, as well as p-adic methods. The second part is dealing with the set of non-negative integers whose base-2 representation contains an even number of 1, called the Thue-Morse set and denoted by T. We study of the minimal automaton of the base-2^p expansions of sets of the form mT+r, where m and p are positive integers and r a remainder between 0 and m−1. In particular, we give the state complexity of such sets. The proposed method is constructive and general for any b-recognizable set of integers. As an application, we get a procedure to decide whether a 2^p-recognizable set given via an automaton is a set of the form mT+r. Finally, in the third part, we study properties of automatic sequences based on Parry and Bertrand numeration systems. We show that Parry-automatic sequences, like Pisot-automatic sequences (and thus in particular like b-automatic sequences) have a sublinear factor complexity. Furthermore, we exhibit a Bertrand-automatic sequence whose factor complexity is quadratic. We also prove that, contrarily to Pisot-automatic sequences, the image of a Parry-automatic sequence under a uniform morphism is not always a Parry-automatic sequence. The same happens for periodic deletion of letters. Last, we give the generalization to multidimensional sequences of a well-known result: a sequence is U-automatic if and only if its U-kernel is finite, U being such that the numeration language is regular

Automatic sequences: from rational bases to trees

The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider these built on languages associated with trees having periodic labeled signatures and, in particular, rational base numeration systems. We obtain two main characterizations of these sequences. The first one is concerned with $r$-block substitutions where $r$ morphisms are applied periodically. In particular, we provide examples of such sequences that are not morphic. The second characterization involves the factors, or subtrees of finite height, of the tree associated with the numeration system and decorated by the terms of the sequence.Comment: 25 pages, 15 figure

Automatic winning shifts

To each one-dimensional subshift $X$, we may associate a winning shift $W(X)$ which arises from a combinatorial game played on the language of $X$. Previously it has been studied what properties of $X$ does $W(X)$ inherit. For example, $X$ and $W(X)$ have the same factor complexity and if $X$ is a sofic subshift, then $W(X)$ is also sofic. In this paper, we develop a notion of automaticity for $W(X)$, that is, we propose what it means that a vector representation of $W(X)$ is accepted by a finite automaton. Let $S$ be an abstract numeration system such that addition with respect to $S$ is a rational relation. Let $X$ be a subshift generated by an $S$-automatic word. We prove that as long as there is a bound on the number of nonzero symbols in configurations of $W(X)$ (which follows from $X$ having sublinear factor complexity), then $W(X)$ is accepted by a finite automaton, which can be effectively constructed from the description of $X$. We provide an explicit automaton when $X$ is generated by certain automatic words such as the Thue-Morse word.Comment: 28 pages, 5 figures, 1 tabl

Arithmetic Dynamics

This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely that they (semi-) conjugate a given continuous (or measure-preserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: (1) Beta-expansions, i.e., the radix expansions in non-integer bases; (2) "Rotational" expansions which arise in the problem of encoding of irrational rotations of the circle; (3) Toral expansions which naturally appear in arithmetic symbolic codings of algebraic toral automorphisms (mostly hyperbolic). We study ergodic-theoretic and probabilistic properties of these expansions and their applications. Besides, in some cases we create "redundant" representations (those whose space of "digits" is a priori larger than necessary) and study their combinatorics.Comment: 45 pages in Latex + 3 figures in ep

Numeration Systems: a Link between Number Theory and Formal Language Theory

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions. We discuss the notion of numeration systems, recognizable sets of integers and automatic sequences. We briefly sketch some results about transcendence related to the representation of real numbers. We conclude with some applications to combinatorial game theory and verification of infinite-state systems and present a list of open problems.Comment: 21 pages, 3 figures, invited talk DLT'201

A Fibonacci analogue of the two's complement numeration system

Using the classic two's complement notation of signed integers, the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers. We introduce a Fibonacci-equivalent of the two's complement notation and we show that addition in this numeration system can be performed by a deterministic finite-state transducer. The result is based on the Berstel adder, which performs addition of the usual Fibonacci representations of nonnegative integers and for which we provide a new constructive proof. Moreover, we characterize the Fibonacci-equivalent of the two's complement notation as an increasing bijection between $\mathbb{Z}$ and a particular language.Comment: v3: 21 pages, 3 figures, 3 tables. v4: 24 pages, added a new section characterizing the Fibonacci's complement numeration system as an increasing bijection. v5: changes after revie

Dynamical Directions in Numeration

International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper