A New Embedding of the 3x + 1 Dynamical system

John Leventides ()
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John Leventides: National and Kapodistrian University of Athens

A chapter in Discrete Mathematics and Applications, 2020, pp 305-337 from Springer

Abstract: 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 ) $$(-1)^{T^k(n)}$$ .

Date: 2020
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-55857-4_12

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DOI: 10.1007/978-3-030-55857-4_12

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