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

This paper establishes comprehensive formulas for counting fixed and exact-periodic points of Chebyshev polynomials modulo odd prime powers by analyzing their lifting behavior from finite fields, distinguishing between split and nonsplit source groups, and characterizing the emergence of "ghost" periodic points through $p$ -adic order analysis.

Original authors: Chatchawan Panraksa, Aram Tangboonduangjit

Published 2026-05-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chatchawan Panraksa, Aram Tangboonduangjit

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magical machine called a Chebyshev Polynomial. You feed it a number, and it spits out a new number. If you keep feeding the output back into the machine, you get a sequence of numbers. Sometimes, the sequence gets stuck in a loop, repeating the same numbers over and over. These loops are called cycles, and the numbers inside them are periodic points.

This paper is a deep dive into what happens when you run this machine not just on normal numbers, but on numbers that wrap around in a specific way, like the hands of a clock. Specifically, the authors are looking at what happens when the "clock" gets bigger and bigger, moving from a simple clock with $p$ hours to a clock with $p^2$ , $p^3$ , and so on (where $p$ is an odd prime number).

Here is the story of their discovery, broken down into simple concepts:

1. The Two-Source Map (The "Mirror" Trick)

The authors realized that to understand how these numbers behave, you can't just look at the numbers themselves. You have to look at them through a special "mirror" or a pair of glasses.

Imagine every number on the clock has a secret twin. The machine treats the number and its twin almost the same way. The authors found that all the numbers on the clock come from two distinct "source groups" (like two different factories).

Factory A produces numbers that behave one way.
Factory B produces numbers that behave another way.

By counting how many "twins" come out of each factory that satisfy a specific condition, they derived a simple formula (a "four-GCD formula") to count exactly how many numbers get stuck in a 1-step loop (fixed points) on the small clock.

2. The "Ghost" Phenomenon (The Magic Trick)

The most exciting part of the paper happens when they move from the small clock ( $p$ ) to a slightly bigger one ( $p^2$ ).

Usually, if a number is stuck in a loop on the small clock, it stays stuck in a loop of the same length on the bigger clock. But sometimes, something weird happens. The authors call these "Ghost Periodic Points."

The Analogy: Imagine a dancer on a small stage who spins in a circle once every second. You move them to a bigger stage. You expect them to keep spinning once a second.
The Ghost: Instead, for most of the dancers, they keep spinning once a second. But for a few special dancers, they suddenly start spinning much slower (or faster) on the big stage. They are "ghosts" because on the small stage, they looked like they were spinning fast, but on the big stage, their true, slower nature is revealed.
The Discovery: The paper proves that for every special dancer (periodic point) on the small clock, there is exactly one dancer who keeps the original speed, and the rest of the dancers (the "ghosts") suddenly change their speed to a new, longer cycle.

3. The "Degenerate" Trouble Spots

Why do these ghosts appear? It happens at specific "trouble spots" on the clock where the machine's behavior is "degenerate" (a bit blurry or unstable).

If the machine is stable at a number, that number lifts uniquely to the bigger clock.
If the machine is unstable (degenerate) at a number, that single number explodes into many possibilities on the bigger clock. The authors figured out exactly how many of these new numbers become ghosts and how many keep their original rhythm.

4. The "Tower" of Loops

When they keep making the clock bigger and bigger (from $p^2$ to $p^3$ , etc.), these ghosts don't just appear once; they form a tower.

Imagine a ladder. At the bottom rung, you have the original loop.
As you go up the ladder (higher powers of $p$ ), the ghosts might split again. Some stay on their current rung, while others jump to a higher rung with an even longer loop.
The authors built a "ladder map" (called a cord tower) that predicts exactly how many points will be on each rung of the ladder and how long their loops will be.

5. The Special Case of "3"

The authors found that their perfect "ladder map" works beautifully for almost all prime numbers (5, 7, 11, etc.). However, the number 3 is a bit of a rebel.

At the number 3, the math gets a little "fuzzy" at the edges. The neat formulas they found for the other numbers break down slightly because the "ghosts" behave differently there. They noted this exception but didn't solve the full puzzle for the number 3, leaving it as a special case for future study.

Summary of the "Magic"

In short, the paper is a guidebook for predicting the behavior of a mathematical machine as you zoom in on its settings.

Counting: They gave a precise recipe to count how many numbers get stuck in loops on a small clock.
Lifting: They explained exactly what happens when you zoom in (move to a bigger clock): most numbers keep their rhythm, but a specific group of "ghosts" changes their rhythm.
Prediction: They created a system to predict exactly how many ghosts there are and what their new rhythm will be, building a "tower" of possibilities as the clock gets infinitely large.

This isn't about building a real machine or a clock; it's about understanding the hidden, rhythmic structure of numbers when they are forced to wrap around in specific ways. The authors have successfully mapped out the "dance floor" for these numbers, showing exactly who stays in the same spot and who becomes a ghost.

Technical Summary: Fixed-Point Lifting and Ghost Periodic Points for Chebyshev Polynomials Modulo Odd Prime Powers

Problem Statement
The paper investigates the dynamics of Chebyshev polynomials $T_n(x)$ acting on the finite rings $\mathbb{Z}/p^k\mathbb{Z}$ for an odd prime $p$ and integer $n \ge 2$ . While the global functional graph structure of these maps over finite fields $\mathbb{F}_p$ is well-established (e.g., Gassert, Qureshi–Panario), explicit arithmetic formulas for fixed-point counts and exact-period point counts over higher prime-power levels $\mathbb{Z}/p^k\mathbb{Z}$ remain less developed, particularly for non-permutation degrees and degenerate residue classes. The primary challenge addressed is the lifting of periodic points from $\mathbb{F}_p$ to $\mathbb{Z}/p^k\mathbb{Z}$ , specifically analyzing cases where standard Hensel lifting fails to be unique (degenerate cases) and identifying "ghost" periodic points—lifts whose exact period strictly increases upon moving to a higher modulus.

Methodology
The authors employ a "two-source" parametrization strategy, leveraging the identity $T_n(\frac{\zeta + \zeta^{-1}}{2}) = \frac{\zeta^n + \zeta^{-n}}{2}$ . This reduces the dynamics of $T_n$ on $\mathbb{F}_p$ to the multiplicative dynamics of $\zeta$ in two source groups: the split group $\mathbb{F}_p^\times$ and the non-split group $\mu_{p+1}$ (roots of unity in $\mathbb{F}_{p^2}$ ).

Key methodological components include:

Source Parametrization: Decomposing fixed and periodic points based on the order $e$ of the source parameter $\zeta$ and the Chebyshev order $\text{corde}(n) = \min\{r \ge 1 : n^r \equiv \pm 1 \pmod e\}$ .
Degeneracy Analysis: Distinguishing between nondegenerate fixed points (where $T_n'(a) \not\equiv 1 \pmod p$ ) and degenerate ones (where $T_n'(a) \equiv 1 \pmod p$ ). The latter require local analysis beyond standard Hensel's lemma.
Local Lifting:
- For nondegenerate points, unique lifting is guaranteed by Hensel's lemma.
- For degenerate non-boundary points, the authors use Teichmüller lifts and analyze the kernel of the norm map in unramified quadratic extensions to count fixed lifts.
- For degenerate boundary points ( $a = \pm 1$ ), the proof relies on $p$ -adic valuation estimates of the polynomial $Q_\varepsilon(x) = (T_n(x)-x)/(x-\varepsilon)$ using $p$ -adic logarithms and exponentials. This analytic step requires the hypothesis $p \ge 5$ to ensure convergence within the strict domain of the $p$ -adic log/exp functions.
Möbius Inversion: Exact-period counts are derived from fixed-point counts of iterates $T_{nj}$ using the classical Möbius inversion formula.

Key Contributions and Results

Fixed-Point Count over $\mathbb{F}_p$ (Theorem 5.1):
The authors derive a closed formula for the number of fixed points $N_1 = |\text{Fix}(T_n, \mathbb{F}_p)|$ expressed as a sum of four greatest common divisors:
$N_1 = \frac{\gcd(n-1, p-1) + \gcd(n+1, p-1) + \gcd(n-1, p+1) + \gcd(n+1, p+1) - 2\delta}{2}$
where $\delta = \gcd(n-1, 2)$ . This formula isolates the length-one cycles from known graph descriptions and organizes them by the split/nonsplit source branches.
Degeneracy Criterion (Proposition 7.1):
A precise criterion is established for when a fixed point is degenerate: $a_0$ is degenerate if and only if $p \mid (n^2 - 1)$ . The number of such degenerate residue classes, denoted $d$ , is explicitly calculated based on the branch ( $n-1$ or $n+1$ ) and the source groups.
Fixed-Point Lifting Formulas (Theorem 7.3):
- First Lift ( $p^2$ ): For any odd prime $p$ , the number of fixed points modulo $p^2$ is $N_2 = N_1 + d(p-1)$ . This holds regardless of whether $T_n$ is a permutation.
- All Levels ( $p^k$ ): For $p \ge 5$ , a closed formula for $N_k$ is provided:
  $N_k = N_1 + d(p^{\min(k-1, s)} - 1)$
  where $s = \nu_p(n^2 - 1)$ . This formula is equivalent to a GCD-based expression involving $(p \pm 1)p^{k-1}$ .
- Limitation: The all-level formula fails for $p=3$ due to boundary valuation estimates at the convergence threshold of the $p$ -adic log/exp functions.
Ghost Periodic Points (Theorem 10.4):
For $p \nmid n$ , the paper identifies a dichotomy in the lifting of periodic residues modulo $p^2$ . Let $t = \text{corde}(n)$ be the base period and $\kappa = \text{corde}_p(n)$ be the lifted source order.
- If $\kappa = t$ , all $p$ lifts retain the base period $t$ .
- If $\kappa > t$ , exactly one lift (the Hensel lift) retains period $t$ , while the remaining $p-1$ lifts are "ghost" periodic points of period $\kappa$ .
Higher-Level Period Towers (Theorem 11.9):
For $p \ge 5$ and $p \nmid n$ , the lifting of periods to arbitrary levels $k$ is governed by a "cord tower" $c_q = \text{corde}_{p^q}(n)$ . The $p^{k-1}$ lifts of a periodic residue are partitioned into sets of points with exact periods corresponding to the distinct values in the tower sequence $c_0, c_1, \dots, c_{k-1}$ .
Exact-Period Counts (Corollary 11.1):
A general Möbius inversion identity is provided to compute the number of points of exact period $r$ over $\mathbb{Z}/p^k\mathbb{Z}$ for any odd prime $p$ . For $p \ge 5$ , this yields explicit closed forms by substituting the fixed-point formulas.

Significance and Scope
The paper claims to provide explicit arithmetic formulas for fixed-point and exact-period point counts that are organized around the two-source parametrization and local lifting behavior. Its significance lies in:

Generality: Addressing arbitrary degrees $n$ , including non-permutation cases and degenerate residue classes, which are often excluded in prior work focused on permutation polynomials (e.g., Li–Lu–Tan–Chen).
Local Precision: Moving beyond global graph descriptions to provide residue-wise lifting rules, specifically characterizing "ghost" orbits where periods bifurcate at the first lift.
Explicit Formulas: Offering closed-form expressions for $N_k$ and $P_r^{(k)}$ that depend on $p$ -adic valuations and GCDs, rather than just asymptotic or structural descriptions.

The authors note that the results for $p \ge 5$ rely on analytic $p$ -adic estimates that do not hold for $p=3$ , and the case $p=2$ is excluded due to the non-invertibility of 2 in the parametrization. The work is presented as an arithmetic reorganization of known finite-field data to facilitate lifting arguments, rather than a new classification of global graphs over $\mathbb{F}_p$ .