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

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 $n\ge2$, every odd prime $p$, and every $k\ge1$. The results apply whether or not $