Collatz on the Dyadic Rationals in [0.5, 1) with Fractals: How Bit Strings Change Their Length Under 3x + 1

Abstract

The reduced Collatz map (Syracuse function) can be stated as “for any odd positive integer x, calculate and then divide by 2 until the result is odd.” We calculate the change in bit string length caused by this map. The result arises from a novel reformulation of the Collatz process “for any fraction in [0.5, 1) with a binary representation of finite length, append If the resulting number is smaller than 2/3 then multiply by 3/2 otherwise multiply by 3/4.” The domain is where strings may shrink versus for may grow. If the Collatz map has non-trivial periodic orbits then they will arise from a fractal that has been added to a map that lacks periodic orbits. If there are an infinite number of such orbits then as their length increases, they will make roughly 0.71 visits to the may-shrink domain for each visit to the may-grow domain.