THE LAP-COUNTING FUNCTION FOR LINEAR MOD ONE TRANSFORMATIONS .3. THE PERIOD OF A MARKOV-CHAIN

Linear mod one transformations are those maps of the unit interval giv en by f(beta,alpha)(x) = beta x + alpha (mod 1), with beta > 1 and 0 l ess than or equal to alpha < 1. The lap-counting function is L-beta,L- alpha(z) = Sigma(n=1)(infinity) L(n)z(n), where L-n essentially counts the number of monotonic pieces of the nth iterate f(beta,alpha)(n). P art I showed that the function L-beta,L-alpha(z) is meromorphic in the unit disk \z\ < 1 and analytic in \z\ < 1/beta, and part II showed th at the singularities of L-beta,L-alpha(z) on the circle \z\ = 1/beta a re contained in the set {(1/beta>exp(2 pi il/N-beta,N-alpha) : 0 less than or equal to 1 < N-beta,N-alpha}, where N-beta,N-alpha is the peri od of the ergodic part of a Markov chain associated to f(beta,alpha). This paper proves that the set of singularities on \z\ = 1/beta is ide ntical to the set {(1/beta)exp(2 pi il/N-beta,N-alpha) : 0 less than o r equal to l < N-beta,N-alpha}. Part II showed that N-beta,N-alpha = 1 for beta > 2, and this paper determines N-beta,N-alpha, in the remain ing cases where 1 < beta less than or equal to 2.