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

Abstract

We study the two-dimensional continued fraction algorithm introduced in [5] and the associated triangle map T , defined on a triangle △ ⊆ R 2 . We introduce a slow version of the triangle map, the map S, which is ergodic with respect to the Lebesgue measure and preserves an infinite Lebesgue-absolutely continuous invariant measure. We show that the two maps T and S share many properties with the classical Gauss and Farey maps on the interval, including an analogue of the weak law of large numbers and of Khinchin's weak law for the digits of the triangle sequence, the expansion associated to T . Finally, we confirm the role of the map S as a two-dimensional version of the Farey map by introducing a complete tree of rational pairs, constructed using the inverse branches of S, in the same way as the Farey tree is generated by the Farey map, and then, equivalently, generated by a generalised mediant operation. CLAUDIO BONANNO, ALESSIO DEL VIGNA, AND SARA MUNDAYWe show that, similarly to the Farey map, S preserves an infinite Lebesgue-absolutely continuous measure and it is ergodic with respect to the Lebesgue measure on △. It follows that the statistical behaviour of summable observables along orbits of S is non-standard. This phenomenon, for the Farey map, makes it impossible to improve Khinchin's weak law for the coefficients of the regular continued fraction expansion to a strong law. However, we are able to exploit certain results from infinite ergodic theory to show that the system generated by S is pointwise dual ergodic and prove a weak law of large numbers for S, from which we obtain an analogue of Khinchin's weak law for the digits of the triangle sequences.The connection between the Gauss and the Farey maps and the regular continued fractions can be studied also through the Farey tree, a binary tree which contains all the rational numbers in (0, 1) (see e.g. [4]). The Farey tree is strongly related to the Farey map, but it can also be defined through the mediant operation on fractions. We recall the definition of the Farey tree and its basic properties in Section 5. Analogously, in this paper we define a tree of rational pairs, first by using a suitable modification of the map S limited to the set of indifferent fixed points, and then by using a generalised mediant operation defined on pairs of rational numbers. We prove that the two trees are in fact identical level by level, and that the tree is complete, that is, it contains every pair of rational numbers in s △. This last result improves on the results of [2], where the authors study different trees generated by the triangle map and its generalisations, but show that none of them are complete.The paper is organised as follows. In Section 2 we recall the definition of the triangle map T and the associated two-dimensional continued fraction algorithm. We also introduce the map S and study its basic ergodic properties. Lastly, we define a dynamical system on an infinite strip, which is isomorphic to the action of S on s △. This isomorphism gives a useful i...

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

“…In this paper we continue the work initiated by the authors with Sara Munday in [7], where we have studied the properties of a tree of rational pairs, here called the Triangular tree, which was introduced as a two-dimensional 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 be found also in the Triangular tree and to construct approximations of real pairs using the tree.…”

“…The analogous of the Farey map in this two-dimensional setting has been introduced in [7]. Let {Γ 0 , Γ 1 } be the partition of s △ such that Γ 0 := △ 0 and Γ 1 := s △ \ Γ 0 .…”

“…Many properties of the map S have been proved in [7]: S is ergodic with respect to the Lebesgue measure, it preserves the infinite Lebesgue-absolutely continuous measure with density 1 xy , and it is pointwise dual ergodic. Finally, the role of the map S as a two-dimensional version of the Farey map is confirmed by the construction of a complete tree of rational pairs, the Triangular tree, by using the inverse branches of S, in the same way as the Farey tree is generated by the Farey map, and then, equivalently, by a generalised mediant operation.…”

“…the main results of [7] 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 [7], 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 [7].…”

“…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 [7], 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 [7]. 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.…”