This paper deals with rational base number systems for p-adic numbers. We mainly focus on the system proposed by Akiyama et al. in 2008, but we also show that this system is in some sense isomorphic to some other rational base number systems by means of finite transducers. We identify the numbers with finite and eventually periodic representations and we also determine the number of representations of a given p-adic number.

We add a sufficient condition for validity of Propo- sition 4.10 in the paper Frougny et al. (2004). This condition is not a necessary one, it is nevertheless convenient, since anyway most of the statements in the paper Frougny et al. (2004) use it.


We study the complexity of the infinite word uβ associated with theRényi expansion of 1 in an irrational base β > 1.When β is the golden ratio, this is the well known Fibonacci word,which is Sturmian, and of complexity C(n) = n + 1.For β such thatdβ(1) = t1t2...tm is finite we provide a simple description ofthe structure of special factors of the word uβ . When tm =1we show thatC(n) = (m - 1)n + 1. In the cases when t1 = t2 = ... tm-1 ort1 > max{t2,...,tm-1 } we show that the first differenceof the complexity function C(n + 1) - C(n ) takes value in{m - 1,m} for every n, and consequently we determine thecomplexity of uβ . We show thatuβ is an Arnoux-Rauzy sequence if and only ifdβ(1) = tt...t1. On the example ofβ = 1 + 2cos(2π/7), solution of X3 = 2X2 + X - 1, we illustratethat the structure of special factors is more complicated fordβ (1) infinite eventually periodic.The complexity for this word is equal to 2n+1.

Two linear numeration systems, with characteristic polynomial equal to the minimal polynomial of two Pisot numbers β and γ respectively, such that β and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other one is computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodic points of a sofic dynamical system associated with some Parry number.

We consider numeration systems where the base is anegative integer, or a complex number which is a root of anegative integer.We give parallel algorithms for addition in these numeration systems,from which we derive on-line algorithms realized by finite automata. A general construction relating addition in base βand addition in base βm is given.Results on addition in base $\beta=\sqrt[m]{b}$ , where b is a relative integer, follow.We also show that addition in base the golden ratiois computable by an on-line finite automaton, but is notparallelizable.

We characterize numbers having finite β-expansions where β belongs to a certain class of Pisot numbers: when the β-expansion of 1 is equal to a1a2…am, where a1≥a2≥…≥am≥1 and when the β-expansion of 1 is equal to t1t2…tm(tm+1)ω where t1≥t2≥…≥tm>tm+1≥1.