Dynamical directions in numeration

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2.5. Questions 3. Canonical numeration systems, β -expansions and shift radix systems 3.1. Canonical numeration systems in number fields 3.2. Generalisations 3.3. On the finiteness property of β -expansions 3.4. Shift radix systems 3.5. Numeration systems defined over finite fields 3.6. Lattice tilings 4. Some sofic fibred numeration systems 4.1. Substitutions and Dumont-Thomas numeration 4.2. Abstract numeration systems 4.3. Rauzy fractals 4.4. The Pisot conjecture 5. G-scales and odometers 5.1. G-scales. Building the odometer 5.2. Carries tree 5.3. Metric properties. Da capo al fine subshifts 5.4. Markov compacta 5.5. Spectral properties 6. Applications 6.1. Additive and multiplicative functions, sum-of-digits functions 6.2. Diophantine approximation 6.3. Computer arithmetics and cryptography 6.4. Mathematical crystallography: Rauzy fractals and quasicrystals acknowledgements References
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1. Introduction 1.1. Origins. Numeration is the art of representation of numbers; primarily natural numbers, then extensions of them - fractions, negative, real, complex numbers, vectors, a.s.o. Numeration systems are algorithmic ways of coding numbers, i.e., essentially a process permitting to code elements of an infinite set with finitely many symbols. For ancient civilisations, numeration was necessary for practical use, commerce, astronomy, etc. Hence numeration systems have been created not only for writing down numbers, but also in order to perform arithmetical operations. Numeration is inherently dynamical, since it is collated with infinity as potentiality, as already asserted by Aristotle1: if I can represent some natural number,
1“The infinite exhibits itself in different ways - in time, in the generations of man, and in the division of magnitudes. For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different. Again, ‘being’ has more than one sense, so that we must not regard the infinite as a ‘this’, such as a man or a horse, but must suppose it to exist in the sense in which we speak of the day or the games as existing things whose being has not come to them like that of a substance, but consists in a process of coming
2000 Mathematics Subject Classification. Primary 37B10; Secondary 11A63, 11J70, 11K55, 11R06, 37A45, 68Q45, 68R15. Key words and phrases. Numeration, fibred systems, symbolic dynamics, odometers, numeration scales, subshifts, f -expansions, β -numeration, sum-of-digits function, abstract number systems, canonical numeration systems, shift radix systems, additive functions, tilings, Rauzy fractals, substitutive dynamical systems. The first author was supported by the Austrian Science Foundation FWF, project S9605, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”. The first threee authors were partially supported by ACINIM “Num´ eration” 2004–154. The fourth author was supported by the FWF grant S9610-Nhe next one? On that score, it is significant that motion (greek δυ ´ναµις ) and infinity are treated together in Aristotle’s work (Physics, third book). Furthermore, the will to deal with arbitrary large numbers requires some kind of invariance of the representation and a recursive algorithm which will be iterated, hence something of a dynamical kind again. In the sequel, we briefly mention the most important historical steps of numeration. We refer to the book of Ifrah [Ifr94] for an amazing amount of information on the subject and additional references. Numeration systems are the ultimate elaboration concerning representation of numbers. Most early representations are only concerned with finitely many numbers, indeed those which are of a practical use. Some primitive civilisations ignored the numeration concept and only had names for cardinals that were immediately perceptible without performing any action of counting, i.e., as anybody can experiment alone, from one to four. For example, the Australian tribe Aranda say “ninta” for one, “tara” for two, “tara-ma-ninta” for three, and “tara-ma-tara” for four. Larger numbers are indeterminate (many, a lot). Many people have developed a representation of natural numbers with fingers, hands or other parts of the human body. Using phalanxes and articulations, it is then possible to represent (or show) numbers up to ten thousand or more. A way of showing numbers up to 1010 just with both hands was implemented in the XVIth century in China (Sua fa tong zong, 1593). Clearly, the choice of base 10 was at the origin of these methods. Other bases were attested as well, like five, twelve, twenty or sixty by Babylonians. However, all representations of common use work with a base. Bases have been developed in Egypt and Mesopotamia, about 5000 years ago. The Egyptians had a special sign for any small power of ten: a vertical stroke for 1, a kind of horseshoe for 10, a spiral for 100, a loto flower for 1000, a finger for 10000, a tadpole for 105 , and a praying man for a million. For 45200, they drew four fingers, five loto flowers and two spirals (hieroglyphic writing). A similar principle was used by Sumerians with base 60. To avoid an over complicated representation, digits (from 1 to 59) were written in base 10. This kind of representation follows an additional logic. A more concise coding has been used by inventing a symbol for each digit from 1 to 9 in base 10. In this modified system, 431 is understood as 4 × 100 + 3 × 10 + 1 × 1 instead of 100 + 100 + 100 + 100 + 10 + 10 + 10 + 1. Etruscans used such a system, as did Hieratic and Demotic handwritings in Egypt. The next crucial step was the invention of positional numeration. It has been discovered independently four times, by Babylonians, in China, by the pre-Columbian Mayas, and in India. However, only Indians had a distinct sign for every digit. Babylonians only had two, for 1 and 10. Therefore, since they used base 60, they represented 157, say, in three blocks: from the left to the