The lap-counting and zeta functions of the tent map

The tent map t(beta) : {[}0, 1] --> {[}0, 1] with parameter 1 B less than or equal to 2 is defined by t(beta) (x) = {beta x 0 less than or equal to x 1/2 beta(1-x) 1/2 less than or equal to x less than or equal to 1. This paper derives formulae for its dynamical zeta and lap-counting functions which exhibit the renormalization structure of such maps. It relates these functions to the centrally symmetric linear mod 1 transformation f(beta)(x)=beta x + 1 - 1/2 beta (mod 1). The singularities of these functions on the circle /z/ = 1/beta are determined explicitly AMS classification scheme numbers: 58F03, 58F20.