The iteration x := ln(x*x) has repelling length p cycles for all p. Consider any periodic cycle x1, x2, ..., xp = x1. Define the (symbolic) "kneading sequence" k1 k2 ... kp over {0,1}* by ki = 0 if xi>=0, ki = 1 if xi<0. The zero kneading sequence is 0 0 ... 0. Then the periodic cycles are in 1-1 correspondence with nonzero necklaces (a length p necklace is the equivalence class of cyclic shifts of a length p string). So there is one fixed point, with kneading necklace 1. Two period 2 cycles with necklaces {01, 11} - necklace 11 also has period 1, so only one strict period 2 cycle. Two strict period 3 cycles, with necklaces {001, 011}. Three strict period 4 cycles, with necklaces {0001, 0011, 0111}. Six strict period 5 cycles, with necklaces {00001, 00011, 00101, 00111, 01011, 01111}. The inverse iteration for a periodic cycle iterates x_(i+1) = +/-sqrt(e^xi), where the iteration selects the negative sqrt branch for kneading symbol ki=1, and nonnegative branch for ki=0. The nonzero kneading sequences give attracting cycles in the inverse iteration, hence repelling cycles in the original iteration. The zero kneading sequence in the inverse iteration diverges to infinity. The inverse iteration attracting cycles all have positive derivative. I'm not sure what this says about critical points or about other chaotic aspects. Anymore with more dynamical systems knowledge in a position to comment? - Scott
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Keith F. Lynch Sent: Thursday, March 21, 2013 5:27 PM To: math-fun@mailman.xmission.com Subject: [math-fun] X^2, ln(X), repeat...
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?
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