A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs

Representation and coding of rational pairs on a triangular tree and Diophantine approximation in $\mathbb R^2$

“…In this section we recall the definitions of the Triangle Map and of its slow version, and the main steps of the construction of the Triangular Tree. For more details we refer the reader to [8,5] where these notions have been introduced.…”

“…In this paper we continue the work initiated by the authors with Sara Munday in [5], where we have studied the properties of a tree of rational pairs, here called the Triangular Tree, which was introduced as a twodimensional version of the well-known Farey tree (or Stern-Brocot tree). The aim of this paper is to show that all the structures of the Farey tree can also be found in the Triangular Tree and to construct approximations of real pairs using the tree.…”

“…In Section 2 we recall the construction of the Farey tree and its main properties based on the continued fraction expansions of real numbers, in particular the {L, R} coding of real numbers which is based on the relations of the Farey tree to the group SL(2, Z). In Section 3 we recall the main results of [5] concerning the Triangular Tree. In particular we recall that it can be constructed in a dynamical way, by using the Triangle Map defined in [8] which acts on the triangle := {(x, y) ∈ R 2 : 1 ≥ x ≥ y > 0} and its slow version S introduced in [5], and in a geometric way, by using the notion of mediant of two pairs of fractions.…”

“…In Section 3 we recall the main results of [5] concerning the Triangular Tree. In particular we recall that it can be constructed in a dynamical way, by using the Triangle Map defined in [8] which acts on the triangle := {(x, y) ∈ R 2 : 1 ≥ x ≥ y > 0} and its slow version S introduced in [5], and in a geometric way, by using the notion of mediant of two pairs of fractions. The existence of these two possible constructions is the first basic property of the Farey tree that has been proved for the Triangular Tree in [5].…”

“…In particular we recall that it can be constructed in a dynamical way, by using the Triangle Map defined in [8] which acts on the triangle := {(x, y) ∈ R 2 : 1 ≥ x ≥ y > 0} and its slow version S introduced in [5], and in a geometric way, by using the notion of mediant of two pairs of fractions. The existence of these two possible constructions is the first basic property of the Farey tree that has been proved for the Triangular Tree in [5]. Afterwards, Section 4 contains some preparatory results on the triangle sequences, that is, the symbolic representation of the orbits of real pairs for the Triangle Map.…”