Fixed points and exact periods of Chebyshev polynomials modulo odd prime powers

Abstract

Passing from a prime modulus to a higher prime power can change both the lengths and the number of cycles of a polynomial map. We determine these changes for Chebyshev polynomials $T_n$ modulo $p^k$, for every degree $n\ge 2$, every odd prime $p$, and every $k\ge 1$. The results apply whether or not $T_n$ permutes the residue classes. We give explicit formulas for the number of fixed points, all possible cycle lengths, and the number of cycles of each length. The fixed-point formula involves four greatest common divisors and a correction specific to $p=3$. We explain this exception by showing how distinct rational fixed points become congruent modulo $3$. When $p\mid n$, each cycle modulo $p$ corresponds to exactly one cycle of the same length modulo every $p^k$. When $p\nmid n$, we determine how many cycles arise from each cycle modulo $p$ and when longer cycles first appear. The proofs combine the classical relation between Chebyshev polynomials and power maps with $p$-adic arithmetic.

23 pages. Revised and reorganized; added explicit fixed-point and exact-period formulas for the ternary boundary cases, with updated comparisons to the literature