Years ago I noticed that if I alternate between hitting the X^2 and the ln(X) buttons on a calculator, the result neither converges nor blows up, but just wanders around, seemingly at random. What, if anything, is known about this sequence? Does it have any fixed points? Any cycles of length N for any N? What is the distribution? The mean value (after the log step) seems to be between 0.15853 and 0.15854. With some values it will blow up. 0 will make it blow up in one step, 1 or -1 will make it blow up in two steps, sqrt(e), -sqrt(e), sqrt(1/e), and -sqrt(1/e) in three steps, etc. The number of such explosive starting values doubles with each step. Is this set of explosive starting values dense, i.e. between any two such values is there always another?