A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs (Q2220381)

A two-dimensional version of a continued fraction introduced by \textit{T. Garrity} [J. Number Theory 88, No. 1, 86--103 (2001; Zbl 1015.11031)] is studied. This defines a map defined on a triangle, and the associated expansions are referred to as triangle sequences. The results here give further parallels between the classical Gauss map and this triangle map. In particular, the role played by the Farey map for the Gauss map finds a parallel in a certain piecewise linear fractional map on the triangle constructed here and shown to preserve an ergodic infinite absolutely continuous measure. Results from infinite ergodic theory are applied to derive weak laws of large numbers and thence an analogue of Khinchin's weak law for the digits of the triangle sequences.