American Institute of Mathematical Sciences

Advanced
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 - A, 2003, 9 (3) : 771-787. doi: 10.3934/dcds.2003.9.771
[1]

Yunhua Zhou. The local $C^1$-density of stable ergodicity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2621-2629. doi: 10.3934/dcds.2013.33.2621
[2]

Quansen Jiu, Zhouping Xin. The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients. Kinetic & Related Models, 2008, 1 (2) : 313-330. doi: 10.3934/krm.2008.1.313
[3]

Yaozhong Hu, David Nualart, Xiaobin Sun, Yingchao Xie. Smoothness of density for stochastic differential equations with Markovian switching. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3615-3631. doi: 10.3934/dcdsb.2018307
[4]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[5]

Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129
[6]

Ping Chen, Daoyuan Fang, Ting Zhang. Free boundary problem for compressible flows with density--dependent viscosity coefficients. Communications on Pure & Applied Analysis, 2011, 10 (2) : 459-478. doi: 10.3934/cpaa.2011.10.459
[7]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005
[8]

Michele Carriero, Antonio Leaci, Franco Tomarelli. Uniform density estimates for Blake & Zisserman functional. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1129-1150. doi: 10.3934/dcds.2011.31.1129
[9]

Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292
[10]

Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001
[11]

Jean-François Couchouron, Mikhail Kamenskii, Paolo Nistri. An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1845-1859. doi: 10.3934/cpaa.2013.12.1845
[12]

Brian Marcus and Selim Tuncel. Powers of positive polynomials and codings of Markov chains onto Bernoulli shifts. Electronic Research Announcements, 1999, 5: 91-101.

[13]

Peter E. Kloeden, Victor Kozyakin. Asymptotic behaviour of random tridiagonal Markov chains in biological applications. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 453-465. doi: 10.3934/dcdsb.2013.18.453
[14]

Jishan Fan, Tohru Ozawa. Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model. Conference Publications, 2011, 2011 (Special) : 400-409. doi: 10.3934/proc.2011.2011.400
[15]

Xian Zhang, Vinesh Nishawala, Martin Ostoja-Starzewski. Anti-plane shear Lamb's problem on random mass density fields with fractal and Hurst effects. Evolution Equations & Control Theory, 2019, 8 (1) : 231-246. doi: 10.3934/eect.2019013
[16]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737
[17]

Jishan Fan, Tohru Ozawa. An approximation model for the density-dependent magnetohydrodynamic equations. Conference Publications, 2013, 2013 (special) : 207-216. doi: 10.3934/proc.2013.2013.207
[18]

Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993
[19]

Enrique R. Pujals. On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 179-226. doi: 10.3934/dcds.2006.16.179
[20]

Rodrigo P. Pacheco, Rafael O. Ruggiero. On C1, β density of metrics without invariant graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 247-261. doi: 10.3934/dcds.2018012

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences