On the combinatorial structure of Rauzy graphs

Let S _m ⁰ be the set of all irreducible permutations of the numbers {1, ??,m} (m ?? 3). We define Rauzy induction mappings a and b acting on the set S _m ⁰ . For a permutation ?? ?? S _m ⁰ , denote by R(??) the orbit of the permutation ?? under the mappings a and b. This orbit can be endowed with the structure of an oriented graph according to the action of the mappings a and b on this set: the edges of this graph belong to one of the two types, a or b. We say that the graph R(??) is a tree composed of cycles if any simple cycle in this graph consists of edges of the same type. An equivalent formulation of this condition is as follows: a dual graph R*(??) of R(??) is a tree. The main result of the paper is as follows: if the graph R(??) of a permutation ?? ?? S _m ⁰ is a tree composed of cycles, then the set R(??) contains a permutation ?? ₀: i ? m + 1 ? i, i = 1, ??,m. The converse result is also proved: the graph R(?? ₀) is a tree composed of cycles; in this case, the structure of the graph is explicitly described.