Abstract :
We consider the following analogue of the 3x + 1 function, [formula], where β > 1 is real, and left ceiling right ceiling is the ceiling function (next largest integer). The case β = image is just the 3x + 1 function. We prove that for almost all β, Tβ decreases iterates on average when 1 < β < 2 and increases iterates on average when β > 2. We find certain values of β where the analogue of the 3x + 1 conjecture has an affirmative answer and other values where it has a negative answer.
