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A New Embedding of the 3x + 1 Dynamical system

In: Discrete Mathematics and Applications

Author

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  • John Leventides

    (National and Kapodistrian University of Athens)

Abstract

The 3x + 1 dynamical system T can be studied via the Collatz graph that depicts the trajectories of T in the set of natural numbers ℕ $$\mathbb {N}$$ *. The study of this graph is problematic as there is no evident structure that can be exploited. We embed this graph and its shifted copies in a new fully binary tree and extend T to a new map T ¯ $$\overline {T}$$ that all its trajectories converge to a single equilibrium. The new graph resembles that of a shift map yet whole Collatz trajectories exist intact within it. This new structure allows the simultaneous study of all important features of the conjecture, such as Collatz sequences, transition of parity vectors and the double indexed sequence ( − 1 ) T k ( n ) $$(-1)^{T^k(n)}$$ .

Suggested Citation

  • John Leventides, 2020. "A New Embedding of the 3x + 1 Dynamical system," Springer Optimization and Its Applications, in: Andrei M. Raigorodskii & Michael Th. Rassias (ed.), Discrete Mathematics and Applications, pages 305-337, Springer.
    • Handle: RePEc:spr:spochp:978-3-030-55857-4_12
    DOI: 10.1007/978-3-030-55857-4_12
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