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

Günther J. Wirsching

doi:10.3934/dcds.2003.9.771

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2003, Volume 9, Issue 3: 771-787. Doi: 10.3934/dcds.2003.9.771

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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

Received: November 30, 2001

Revised: September 30, 2002

Published: April 2003

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Abstract

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.

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Mathematics Subject Classification: 39A10, 11B85, 60J05.

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