[math-fun] Golden ratio things

I'd like to understand better the connections between the following three things: (1) Zeckendorf representation of positive integers. (2) Beta-expansion of positive real numbers where beta is the golden ratio. (3) Symbolic encoding of points in the torus R^2 / Z^2 = (R/Z)^2 under the action of the toral automorphism associated with the matrix (1 1) (1 0) Can any of you enlighten me (or point me toward sources of enlightenment)? Thanks, Jim Propp