We determine the subword complexity of the characteristic functions of a two-parameter family fA n g 1 n=1 of infinite sequences which are associated with the winning strategies for a family of 2-player games. A special case of the family has the form A n = bnffc for all n 2 Z?0 , where ff is a fixed positive irrational number. The characteristic functions of such sequences have been shown to have subword complexity n + 1. We show that every sequence in the extended family has subword complexity O(n). 1 Introduction Denote by Z0 and Z?0 the set of nonnegative integers and positive integers respectively. Given two heaps of finitely many tokens, we define a 2-player heap game as follows. There are two types of moves: 1. Remove any positive number of tokens from a single heap. 2. Remove k ? 0 tokens from one heap and l ? 0 from the other. Here k and l are constrained by the condition: 0 ! k l ! sk + t, where s and t are predetermined positive integers. The player who reaches a stat...

Abstract. For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the n-th word of a genealogically ordered language L onto the (n+1)-th word of L. We show that, if the ratio of the number of elements of length n +1overthenumber of elements of length n of the language has a limit β>1, then the amortized cost of the successor function is equal to β/(β − 1). From this, we deduce the value of the amortized cost for several classes of numeration systems (integer base systems, canonical numeration systems associated with a Parry number, abstract numeration systems built on a rational language, and rational base numeration systems). 1

this paper. The fi-integers are defined via a numeration system with base fi, see below. In

Abstract: The splitting method was defined by the author in [Margenstern 2002a, Margenstern 2002d]. It is at the basis of the notion of combinatoric tilings. As a consequence of this notion, there is a recurrence sequence which allows us to compute the number of tiles which are at a fixed distance from a given tile. A polynomial is attached to the sequence as well as a language which can be used for implementing cellular automata on the tiling. The goal of this paper is to prove that the tiling of hyperbolic 4D space is combinatoric. We give here the corresponding polynomial and, as the first consequence, the language of the splitting is not regular, as it is the case in the tiling of hyperbolic 3D space by rectangular dodecahedra which is also combinatoric. 1 Key Words: cellular automata, hyperbolic plane

We establish quantitative refinements of recent results on the occurrence of blocks in digital expansions. Furthermore we extend these results to linear numeration systems.

Abstract. Odometers or “adding machines ” are usually introduced in the context of positional numeration systems built on a strictly increasing sequence of integers. We generalize this notion to systems defined on an arbitrary infinite regular language. In this latter situation, if (A, <) is a totally ordered alphabet, then enumerating the words of a regular language L over A with respect to the induced genealogical ordering gives a one-to-one correspondence between N and L. In this general setting, the odometer is not defined on a set of sequences of digits but on a set of pairs of sequences where the first (resp. the second) component of the pair is an infinite word over A (resp. an infinite sequence of states of the minimal automaton of L). We study some properties of the odometer like continuity, injectivity, surjectivity, minimality,... We then study some particular cases: we show the equivalence of this new function with the classical odometer built upon a sequence of integers whenever the set of greedy representations of all the integers is a regular language; we also consider substitution numeration systems as well as the connection with β-numerations. 1.

Let fi be a real number ? 1. The digit set conversion between real numbers represented in fixed base fi is shown to be computable by an on-line algorithm, and thus is a continuous function. When fi is a Pisot number the digit set conversion is computable by an on-line finite automaton. 1 Introduction In computer arithmetic, on-line computation consists of performing arithmetic operations in Most Significant Digit First (MSDF) mode, digit serially after a certain latency delay [8]. This allows the pipelining of different operations such as addition, multiplication and division. It is also appropriate for the processing of real numbers having infinite expansions: it is well known that when multiplying two real numbers, only the left part of the result is significant. To be able to perform on-line addition, it is necessary to use a redundant number system (see [19], [8]). On the other hand, a function is computable by a finite automaton if it needs only a finite auxiliary storage me...

This paper extends some results of Allouche and Shallit for q-regular sequences to numeration systems in algebraic number fields and to linear numeration systems. We also construct automata that perform addition and multiplication by a fixed number. 1 Introduction A sequence is called q-automatic if its n-th term can be generated by a finite state machine from the q-ary digits of n. The concept of automatic sequences was introduced in 1969 and 1972 by Cobham [8, 9]. In 1979 Christol [6] (see also Christol, Kamae, Mend`es France and Rauzy [7]) discovered a nice arithmetic property of automatic sequences: a sequence with values in a finite field of characteristic p is p-automatic if and only if the corresponding power series is algebraic over the field of rational functions over this finite field. A brief survey on this subject is given in [2], see also [10]. Some generalizations of this concept were studied in [27, 23, 24, 3], see also the survey [1]. An automatic sequence has to t...

Aim: Present a systematic development of part of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Isolate the various difficulties separating hard from easy games, and attack them individually. Presentation: Informal; examples of games sampled from various levels illustrate the theory, with emphasis on formulating and motivating new and old research problems.

We study exactness and maximal automorphic factors of C 3 unimodal maps of the interval. We show that for a large class of innitely renormalizable maps, the maximal automorphic factor is an odometer with an ergodic nonsingular measure. We give conditions under which maps with absorbing Cantor sets have an irrational rotation on a circle as a maximal automorphic factor, as well as giving exact examples of this type. We also prove that every C 3 S-unimodal map with no attractor is exact with respect to Lebesgue measure. Additional results about measurable attractors in locally compact metric spaces are given.