Asymmetric Tent Map Expansions I - Eventually Periodic Points.

The family of asymmetric tent maps T sub alpha : [0, 1] -> [0, 1] for alpha > 1 is defined by T sub alpha (x) = left {alpha x for 0 = x = 1 over alpha, alpha over alpha-1, alpha over alpha-1 (1-x) for 1 over alpha = x = 1. Such mappings form an interesting family of discrete dynamical systems, and can be used to give expansions of real numbers in [0, 1] analogous to the decimal expansion. Let rPer (T sub alpha) denote the set of eventually periodic points under iteration of T sub alpha. One always has rPer (T sub alpha) (= dQ (alpha) cap [0, 1]; this paper studies when equality holds. Necessary conditions for equality are that alpha and alpha over alpha - 1 be algebraic integers and that for all embeddings sigma : Q (alpha) -> C with sigma (alpha) = alpha, one has min (| sigma (alpha)|, | sigma (alpha over {alpha - 1})|) 1. Sufficient conditions for rPer (T sub alpha) = Q (alpha) inter [0,1] consist of these conditions plus max (| sigma (alpha) |, | sigma (alpha over {alpha - 1}) | ) 1, i.e., whenever alpha and alpha over {alpha - 1} are both Pisot numbers. We call such numbers special Pisot numbers, find eleven of them, and prove that there are finitely many special Pisot numbers. Certain other alpha are conjectured to have rPer (T sub alpha) = Q (alpha) inter [0, 1], e.g., the real root alpha of X sup 5 - X sup 3 - 1 = 0.