Clark has defined the notion of n-avoidance basis which contains the avoidable formulas with at most n variables that are closest to be unavoidable in some sense. The family C i of circular formulas is such that C 1 = AA, C 2 = ABA.BAB, C 3 = ABCA.BCAB.CABC and so on. For every i n, the n-avoidance basis contains C i . Clark showed that the avoidability index of every circular formula and of every formula in the 3-avoidance basis (and thus of every avoidable formula containing at most 3 variables) is at most 4. We determine exactly the avoidability index of these formulas.

In this paper we present a construction of Kari-Culik aperiodic tile set -the smallest known until now. With the help of this construction, we prove that this tileset has positive entropy. We also explain why this result was not expected. * supported by ANR project EMC NT09 555297

A word is quasiperiodic (or coverable) if it can be covered by occurrences of another finite word, called its quasiperiod. This notion was previously studied in the domains of text algorithms and combinatorics of right infinite words. We extend several results to two dimensions. We also characterize all rectangular words that cover non-periodic two-dimensional infinite words. Then we focus on two-dimensional words with infinitely many quasiperiods. We show that such words have zero entropy. However, contrarily to the one-dimensional case, they may not be uniformly recurrent.