# Numeration and Substitution

Numeration refers to the arithmetic and symbolic representation of mathematical objects, such as the decimal expansion of a real number. Although we always had ten fingers, decimal expansions are by no means the only, or the oldest, or the most efficient way of representation, as it comes to data storage or particular operations we want to carry out. For example, continued fractions, dating back to Greek mathematics, are the foremost method for achieving rational approximations of real numbers. Their importance has not dwindled, due to applications in mechanics (resonance), mathematical analysis (normal forms, small divisor problems, KAM theory) and in complex dynamics. Variations of continued fraction continue to be developed and refined. Also in higher dimensions, the study of continued fractions algorithms (such as Brun, Jacobi-Perron, Selmer and many others) with special emphasis on their dynamical, Diophantine and combinatorial behaviour are important objects of current investigations.

On the other hand, digit systems have to be generalized in various ways and objects such as *β*-expansions, Cantor expansions, canonical number systems, and shift radix systems show a great variety of interesting dynamical, geometric, and combinatorial properties and have even links to complex dynamics recently.

It remains a fascinating fact how many similarities exist between algorithmic approaches to fractal geometry, tiling spaces, complexity theory of symbolic dynamics, piecewise isometries such as interval exchange transformations, but also many problems in combinatorics. In this sense, the theory of numeration systems remains a unifying link between many areas of mathematics and computer science.