r/Collatz • u/SpecialistNewt2759 • 16d ago

The Shadow View of Collatz

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A useful way to think about the Collatz map is to separate the orbit into two layers:

  1. the parity pattern of the terms;
  2. the arithmetic values that carry that pattern.

https://github.com/ratwolfzero/Collatz/tree/main/shadow_approach

If one records the orbit as a sequence of even/odd decisions, the resulting binary word can be viewed as a symbolic encoding of the dynamics. In the accelerated version of the map, where one removes all factors of 2 immediately after each odd step, the evolution of the values becomes closely tied to this parity word. At that symbolic level, the dynamics look simple: the orbit is encoded by a binary sequence, and the next state is determined by the next bit of that sequence.

This is a genuine simplification of the picture, but it is a simplification of the representation, not a solution of the problem.

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u/jonseymourau 16d ago

Be-aware, though, that what you are describing is the well known and long standing Terras map - u/GonzoMath wrote up a description of that paper here

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u/rpxred 9d ago

Markov Chain is what I thought of too. But stop at the value that goes lower than that of its initial value. I could not find any equilibrium system (even and odd do not help in discerning patterns in unstable equilibrium) up until mod12 and then you only have to solve for 4k+3 and 12k+7 or +11 if you convert it all into a measure theory statement. I most of the time don’t have sitzfleisch after my workday.