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May  2003, 9(3): 771-787. doi: 10.3934/dcds.2003.9.771

On the problem of positive predecessor density in $3n+1$ dynamics

Günther J. Wirsching 1,

1. 

Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt, D-85071 Eichstätt, Germany

Received  December 2001 Revised  October 2002 Published  February 2003

The $3n+1$ function is given by $T(n)=n/2$ for $n$ even, $T(n)=(3n+1)/2$ for $n$ odd. Given a positive integer $a$, another number $b$ is a called a predecessor of $a$ if some iterate $T^\nu(b)$ equals $a$. Here some ideas are described which may lead to a proof showing that the set of predecessors of $a$ has positive lower asymptotic density, for any positive integer $a\ne 0 $ mod 3. Three unbridged gaps in the argument are formulated as conjectures.
Keywords: predecessor density, $3x+1$ problem, strongly stable Markov chains, positive asymptotic density, functional differential equations., Collatz problem.
Mathematics Subject Classification: 39A10, 11B85, 60J0.
Citation: Günther J. Wirsching. On the problem of positive predecessor density in $3n+1$ dynamics. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 771-787. doi: 10.3934/dcds.2003.9.771
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