Let M({nk}k≥1, {ck}k≥1) be the collection of homogeneous Moran sets deter-mined by {nk}k≥1 and {ck}k≥1, where {nk}k≥1 is a sequence of positive integers and {ck}k≥1 a sequence of positive numbers. In this paper, we determine the maximal and minimal values of Hausdorff dimensions for elements in M. We also prove that for any value s between the maximal and minimal values, there exists an element in M({nk}k≥1, {ck}k≥1) such that its Hausdorff dimension is equal to its. The same results hold for Packing dimension. In the meantime, some other properties of homogeneous Moran sets are discussed.

Let F be the free group generated by the letters a and b. let φ be a homomorphism from P to SL2(￠). Lei χ=Trφ(a), y=Trφ(b), z=Trφ(ab) and let w ε F, then Trφ(w) will be a polynomial of variables x, y. z. The main aim of this note is to determine the leading term of Trφ(w). Let σ be an endomorphlsm of F, then σ witl indite a mapping φσ:(Z[x, y, z])3→(Z[x, y.=1)3 which is called the trace mapping associated with σ. We wig establish the recurrence relations of the reading terms and degrees of the iterations of φσ, Moreover, some typical examples are studied.

Nous donnons un procede permettant d'obtenir de nombreuses relations entre les traces Clans des algebres de matrices Cette 6rude fait suite a de nombreuses autres tant cn mathematlques ([1], [3], [7]a. [11], [13]) qu'cn physique (par cxemple, [2], [4], [5], [6]) ou ces relations permettent d'etudier le spectre de certains operateurs de Schredinger disrets, a potentiel defini par un automate fini, intervenant dans la modelisation des quasi-cristaux decouverts par Shechtman et al [12].

The combinatorial properties of the Fibonacci infinite word are of great interest in some aspects of mathematics and physics, such as number theory, fractal geometry, formal language, computational complexity, quasicrystals etc. In this note, we introduce the singular words of the Fibonacci infinite word and discuss their properties. We establish two decompositions of the Fibonacci word in singular words and their consequences. By using these results, we discuss the local isomorphism of the Fibonacci word and the overlap properties of the factors. Moreover, we also give new proofs for the results on special words and the power of the factors.