The autoconjugacy of a generalized Collatz map

Many of the 2-adic properties of the 3x+1 map generalize to the analogous mx+r map, where m and r are odd integers. We introduce the corresponding autoconjugacy map, prove some simple properties of it and make some further conjectures in the general setting, including weak versions of the periodicity and divergent trajectories conjectures

Endomorphisms of the shift dynamical system, discrete derivatives, and applications

All continuous endomorphisms f[subscript ∞] of the shift dynamical system S on the 2-adic integers Z[subscript 2] are induced by some f : B[subscript n]→{0,1}, where n is a positive integer, B[subscript n] is the set of n-blocks over {0, 1}, and f[subscript ∞](x)=y[subscript 0]y[subscript 1]y[subscript 2]…f[subscript ∞](x) = y[subscript 0]y[subscript 1]y[subscript 2]… where for all i∈N, yi = f(x[subscript i]x[subscript i+1]…x[subscript i+n−1]). Define D:Z[subscript 2]→Z[subscript 2] to be the endomorphism of S induced by the map {(00,0),(01,1),(10,1),(11,0)} and V:Z[subscript 2]→Z[subscript 2] by V(x)=−1−x. We prove that D, V∘DV∘D, S, and V∘S are conjugate to S and are the only continuous endomorphisms of S whose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3x+1 function T:Z[subscript 2]→Z[subscript 2] to a parity-neutral dynamical system. We also construct a conjugacy R from D to T. We apply these results to establish that, in order to prove the 3x+1 conjecture, it suffices to show that for any m∈Z[superscript +], there exists some n∈N such that R[superscript −1](m) has binary representation of the form [bar over x[subscript 0]x[subscript 1]…x[subscript 2n−1]] or [bar over x[subscript 0]x[subscript 1]x[subscript 2]…x[subscript 2n]]

The Collatz conjecture and De Bruijn graphs

We study variants of the well-known Collatz graph, by considering the action of the 3n+1 function on congruence classes. For moduli equal to powers of 2, these graphs are shown to be isomorphic to binary De Bruijn graphs. Unlike the Collatz graph, these graphs are very structured, and have several interesting properties. We then look at a natural generalization of these finite graphs to the 2-adic integers, and show that the isomorphism between these infinite graphs is exactly the conjugacy map previously studied by Bernstein and Lagarias. Finally, we show that for generalizations of the 3n+1 function, we get similar relations with 2-adic and p-adic De Bruijn graphs.Comment: 9 pages, 8 figure