Fixed-point lifting and ghost periodic points for Chebyshev polynomials modulo odd prime powers

TL;DR

This paper analyzes fixed and periodic points of Chebyshev polynomials over odd prime power rings, providing explicit formulas and lifting methods, with special attention to prime 3 and higher lifts.

Contribution

It introduces new formulas for counting fixed and periodic points of Chebyshev polynomials modulo odd prime powers, including prime-specific boundary cases and lifting techniques.

Findings

Explicit fixed-point formulas over _p and _{p^k}

Counting of periodic points using Chebyshev order and Mf6bius inversion

Lifting behavior of periodic points modulo prime powers, including ghost points

Abstract

Let $p$ be an odd prime, let $n\ge2$, and let the $n$th Chebyshev polynomial $T_n$ act on $\Z/p^k\Z$. We count fixed and exact-periodic points, allowing non-permutation degrees, and organize the finite-field formulas by the two source groups needed for prime-power lifting. Over $\Fp$ we record the four-GCD fixed-point formula \[ N_1=\frac{\gcd(n-1,p-1)+\gcd(n+1,p-1)+\gcd(n-1,p+1)+\gcd(n+1,p+1)-2\delta}{2}, \] where $\delta=\gcd(n-1,2)$. The proof separates split and nonsplit source groups for $a=(\zeta+\zeta^{-1})/2$ and counts degenerate fixed residues branch-wise. For every odd $p$, \[ N_2=N_1+d(p-1). \] Here $d$ denotes the number of fixed residue classes $a\in\Fp$ for which \(T_n'(a)\equiv1\pmod p\). For $p\ge5$ and all $k\ge1$, \[ N_k=N_1+d\bigl(p^{\min(k-1,\nup(n^2-1))}-1\bigr). \] This all-level formula does not extend unchanged to $p=3$, where boundary $p$-adic estimates at $a=\pm1$ can fail; the first-lift formula remains valid. For periods, we use the Chebyshev order \[ \cord_e(n)=\min\{r\ge1:n^r\equiv\pm1\pmod e\}. \] A source-order-$e$ point is periodic over $\Fp$ exactly when $\gcd(n,e)=1$, with period $\cord_e(n)$. M\"obius inversion for the iterates $T_{n^j}$ gives exact-period point counts over $\Z/p^k\Z$ for all odd $p$; for $p\ge5$, the all-level fixed-point formula gives closed forms. When $p\nmid n$, orbitwise lifting modulo $p^2$ gives either full period retention or one Hensel lift plus ghost periodic points of period $\cord_{ep}(n)$. For $p\ge5$, higher lifts above a periodic residue are governed by the tower $\cord_{ep^q}(n)$.