We study the analogue of the Collatz map in the polynomial ring Fp[x], for any prime number p, and the corresponding dynamical system. We show that every f∈Fp[x] is eventually periodic in this system, in a quadratic number of iterations in deg(f), and describe explicitly all corresponding cycles. This extends a result of Hicks, Mullen, Yucas and Zavislak, who studied the case p=2. We also study the Collatz map in the formal power series ring Fp[[x]], observe that in Fp[[x]] all but countably many power series generate divergent trajectories via iterations of this map, and characterize those power series that are eventually periodic.
Bibliographical noteFunding Information:
The first author is grateful for the support of the Israel Science Foundation (grant no. 353/21 ). We also thank the anonymous referee for his/her feedback and comments.
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- Collatz problem
- Finite fields