ASYMMETRIC TENT MAP EXPANSIONS .1. EVENTUALLY PERIODIC POINTS

The family of asymmetric tent maps T(alpha): [0, 1] --> [0, 1] for alpha > 1 is [GRAPHICS] Such mappings form a well-studied family of discrete dynamical systems, which can be used to give expansions of real numbers in [0, 1] analogous to the decimal expansion. Let Per(T(alpha)) denote the set of eventually periodic points under iteration of T(alpha). One always has Per(T(alpha)) subset-or-equal-to Q(alpha) and [0, 1]; this paper studies when equality holds. A sufficient condition for Per (T(alpha) = Q(alpha) and [0, 1] is that both alpha and alpha/(alpha - 1) be Pisot numbers. We call such numbers special Pisot numbers, and exhibit eleven of them. We prove that there are only finitely many special Pisot numbers. Some necessary conditions are given for Per (T(alpha)) = Q(alpha) and [0, 1], and we conjecture that the set of a for which this equality holds includes some alpha which are not special Pisot numbers, in particular the real root of X5-X3-1 = 0