On some random variables and mappings related to the natural extension of the Gauss shift

Abstract

We compare the dynamics of the two-dimensional extensions W and Y of the Gauss and Farey shifts, respectively. We discuss some consequences of the fact that Y is conjugate to the Szekeres diophantine approximation mapping recently studied by Lagarias and Pollington. Using the ergodic properties of W, we determine the limiting distributions of certain Z-indexed sequences of random variables of the form X n = ( f o W n )( x, y) where f is a measurable real function on the domain of W. We apply this to give the solutions to several metrical problems in the lattice plane. Finally, we consider the spectra of values X( x, y) = inf{ X n : n ϵ Z} and X′( x, y) = lim inf n → ∞ X n and comment on some interrelations with familiar problems in the Geometry of Numbers.