Generalized inverses and the total stopping times of collatz sequences |
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Authors: | Roger E. Zarnowski |
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Affiliation: | 1. Department of Mathematics , Angelo State University , San Angelo, Texasroger.zarnowski@angelo |
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Abstract: | Let f(n) be defined on the set N is even, and f(n)=3n+1 if nie: is odd. A well-known conjecture in number theory asserts that for every n the sequence of iterates eventually reaches the cycle (4,2,1). We recast the conjecture in terms of a denumerable Markov chain with transition matrix P. Assuming that (4,2,1) is the only cycle, but allowing for the possibility of unbounded trajectories, we establish the complete structure of a particular generalized inverse X of I?P and show that the entries of X describe the trajectories and "total stopping times" of integers n. Moreover, the infinite matrix X satisfies properties which, in the case of finite matrices, are the defining properties of the unique group generalized inverse (I?P)#. The result extends to dynamical systems on ? consisting of points that are fixed, eventually fixed, or have unbounded trajectories. As a consequence, we obtain a generalized inverse that encodes the dynamics of such systems, and for cases in which known general criteria for the existence of (I?P)# do not apply. |
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Keywords: | Group inverse Drazin inverse 3n+l problem 3x+l problem Collatz sequence |
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