ON A GENERALIZATION OF THE 3X+1 PROBLEM

We consider the following analogue of the 3x + 1 function, [GRAPHICS] where beta > 1 is real, and [] is the ceiling function (next largest integer). The case beta = 3/2 is just the 3x + 1 function. We prove that for almost all beta, T-beta decreases iterates on average when 1 < beta < 2 and increases iterates on average when beta > 2, We find certain values of beta where the analogue of the 3x + 1 conjecture has an affirmative answer and other values where it has a negative answer. (C) 1995 Academic Press, Inc.