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I recently read with great interest the 1999 paper :

"The 3n+1-Problem and Holomorphic Dynamics Simon Letherman, Dierk Schleicher, and Reg Wood" Experimental Mathematics 8: 3 (1999), 241–252.

I found the conjecture relating to the absence of wandering domains implying that all (integer) orbits are eventually periodic very interesting.

My question : After a quick search looking for more recent results or generalisations in this line and not finding anything, does anyone here know of more recent results in this line of approach to the 3n+1 problem ?

If so a link to a pdf would be great ! I'm also interested in any papers or results you may know of that involve further abstractions e.g. to algebraic geometry or algebraic topology or connections ...

Again, links to any more recent papers (last few years) or results would be great !

PS : I already reviewed "Jeffrey C. Lagarias (2006). "The 3x + 1 problem: An annotated bibliography, II (2000–)" were I originally came across this paper.

Should you the reader or any of your colleagues have worked on this or related approaches using ideas from e.g. algebraic geometry or algebraic topology, ... it would be very interesting to hear of your experiences / thoughts

Thanks in advance, Aethelred

aethelred_the_unwashed
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  • The place to start with all Collatz questions is the book by Lagarias. – Gerry Myerson Jul 31 '14 at 13:13
  • thanks, gerry that's were I came across this particular approach and paper. I'm looking for any more recent developments if you may know of them. i.e. specifically along this line ? cheers, – aethelred_the_unwashed Jul 31 '14 at 13:17
  • If you have access to a university library, and they have access to MathSciNet (Math Reviews online), then you can find the review of that paper, and see whether any later papers have cited it. – Gerry Myerson Jul 31 '14 at 13:19
  • I work independently at the moment so that's not something I can try soon, that's why I'm asking the folks here ;-) thanks again though Gerry ! – aethelred_the_unwashed Jul 31 '14 at 13:23

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The review of the paper links to the review of this one:

Dumont, Jeffrey P.; Reiter, Clifford A.; Real dynamics of a 3-power extension of the $3x+1$ function, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10 (2003), no. 6, 875–893, MR2008752 (2005e:37099).

The review of the Dumont-Reiter paper (written by Lagarias) says, "Related work extending the $3x+1$ iteration to other holomorphic functions includes that of M. Chamberland [Dynam. Contin. Discrete Impuls. Systems 2 (1996), no. 4, 495–509; MR1431647 (97m:11028)] and S. Letherman, D. Schleicher and R. M. W. Wood [Experiment. Math. 8 (1999), no. 3, 241–251; MR1724157 (2000g:37049)]," so maybe the Chamberland paper is worth a look.

All of these papers are in the references of Konstadinidis, Pavlos B., The real $3x+1$ problem, Acta Arith. 122 (2006), no. 1, 35–44, MR2217321 (2007c:11029), so that one might interest you, as well.

Gerry Myerson
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  • another area I looked at was to examine dynamical systems with similar properties that are better understood try to establish topological conjugacy ... just looking at different approaches ... – aethelred_the_unwashed Aug 01 '14 at 12:04