Fixed-point lifting and ghost periodic points for Chebyshev polynomials modulo odd prime powers
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| Natura: | Preprint |
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2026
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| _version_ | 1866915983651766272 |
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| author | Panraksa, Chatchawan Tangboonduangjit, Aram |
| author_facet | Panraksa, Chatchawan Tangboonduangjit, Aram |
| contents | 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 \[
N_1=\frac{\gcd(n-1,p-1)+\gcd(n+1,p-1)+\gcd(n-1,p+1)+\gcd(n+1,p+1)-2δ}{2}, \] where $δ=\gcd(n-1,2)$. The proof separates split and nonsplit source groups for $a=(ζ+ζ^{-1})/2$ 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 $\cord_e(n)$. Möbius 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 $\cord_{ep^q}(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04417 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fixed-point lifting and ghost periodic points for Chebyshev polynomials modulo odd prime powers Panraksa, Chatchawan Tangboonduangjit, Aram Number Theory Dynamical Systems Primary 11T06, 37P05, Secondary 11S82, 37P25 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 \[ N_1=\frac{\gcd(n-1,p-1)+\gcd(n+1,p-1)+\gcd(n-1,p+1)+\gcd(n+1,p+1)-2δ}{2}, \] where $δ=\gcd(n-1,2)$. The proof separates split and nonsplit source groups for $a=(ζ+ζ^{-1})/2$ 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 $\cord_e(n)$. Möbius 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 $\cord_{ep^q}(n)$. |
| title | Fixed-point lifting and ghost periodic points for Chebyshev polynomials modulo odd prime powers |
| topic | Number Theory Dynamical Systems Primary 11T06, 37P05, Secondary 11S82, 37P25 |
| url | https://arxiv.org/abs/2605.04417 |