Abstract. This paper deals with rational base number systems for p-adic numbers. We mainly focus on the system proposed by Akiyama, Frougny, and Sakarovitch in 2008, but we also show that this system is in some sense isomorphic to other ones. 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 study bispecial factors in fixed points of morphisms. In particular, we propose a simple method of how to find all bispecial words of non-pushy circular D0L-systems. This method can be formulated as an algorithm. Moreover, we prove that non-pushy circular D0L-systems are exactly those with finite critical exponent.Comment: 18 pages, 5 figure

Abstract. We describe a simple algorithm which, for a given D0L-system, returns all words v such that v k is a factor of the language of the system for all k. This algorithm can be used to decide whether a D0L-system is repetitive.

Any infinite uniformly recurrent word u can be written as concatenation of a finite number of return words to a chosen prefix w of u. Ordering of the return words to w in this concatenation is coded by derivated word du(w). In 1998, Durand proved that a fixed point u of a primitive morphism has only finitely many derivated words du(w) and each derivated word du(w) is fixed by a primitive morphism as well. In our article we focus on Sturmian words fixed by a primitive morphism. We provide an algorithm which to a given Sturmian morphism ψ lists the morphisms fixing the derivated words of the Sturmian word u = ψ(u). We provide a sharp upper bound on length of the list.

The factor complexity of the infinite word u β canonically associated to a non-simple Parry number β is studied. Our approach is based on the notion of special factors introduced by Berstel and Cassaigne. At first, we give a handy method for determining infinite left special branches; this method is applicable to a broad class of infinite words which are fixed points of a primitive substitution. In the second part of the article, we focus on infinite words u β only. To complete the description of its special factors, we define and study (a, b)-maximal left special factors. This enables us to characterize non-simple Parry numbers β for which the word u β has affine complexity.