Periodic Points of Power Maps in Finite Matrix Groups and Algebras

1. Problem Statement

The paper investigates the dynamics of power maps $f(x) = x^L$ on finite algebraic structures, specifically:

• The full matrix algebra $M_n(q)$ over a finite field $\mathbb{F}_q$.
• The General Linear Group $GL_n(q)$.
• The Symplectic Group $Sp_{2n}(q)$.
• The Unitary Group $U_n(q)$.

Here, $q$ is a power of a prime $p$, and $L$ is a prime number such that $L \neq 2$ and $L \mid (q-1)$.

The central question is to determine the proportion of periodic points under the map $x \mapsto x^L$. A point $\alpha$ is periodic if there exists an integer $k > 0$ such that $f^k(\alpha) = \alpha$. The author aims to compute the limit of the ratio of the number of periodic points to the total size of the group/algebra as $q \to \infty$, subject to a constraint on the $L$-adic valuation of $q-1$.

Specifically, the paper seeks to evaluate:
$$\lim_{\substack{q \to \infty \\ v_L(q-1)=c}} \frac{|\text{Per}(x^L, G)|}{|G|}$$
where $G$ represents one of the aforementioned groups or algebras, and $v_L$ denotes the $L$-adic valuation.

2. Methodology

The author employs a combination of arithmetic dynamics, finite field theory, and representation theory (specifically the classification of conjugacy classes).

• Characterization of Periodic Points:

  • Using Proposition 3.1, the paper establishes that a matrix $A \in M_n(q)$ is periodic under $x \mapsto x^L$ if and only if:
    1. All eigenvalues of $A$ are periodic in the multiplicative group of the algebraic closure.
    2. The Jordan canonical form of $A$ contains no non-trivial nilpotent blocks (i.e., the matrix is semisimple).
  • This reduces the problem to counting elements with specific eigenvalue properties and diagonalizable structures.

• Counting Irreducible Polynomials:

  • The core combinatorial tool involves counting monic irreducible polynomials over $\mathbb{F}_q$ (and $\mathbb{F}_{q^2}$ for unitary groups) whose roots satisfy specific power conditions (e.g., $\alpha^{e_n} = 1$).
  • Lemmas 2.1, 2.2, and 2.3 provide explicit formulas for the number of such polynomials using the Möbius inversion formula and properties of $L$-adic valuations.
  • Special attention is paid to self-reciprocal and self-conjugate polynomials required for the classification of Symplectic and Unitary groups.

• Asymptotic Analysis:

  • The author analyzes the size of conjugacy classes. It is shown (Proposition 3.5) that as $q \to \infty$, the contribution of matrices with eigenvalue 0 or non-trivial nilpotent parts to the total count becomes negligible (polynomials of degree strictly less than the dimension of the space).
  • Therefore, the limiting proportion is determined entirely by regular semisimple invertible elements.

• Partition Theory:

  • The counting is organized by partitions of the dimension $n$ (or $\ell$). The structure of the conjugacy classes corresponds to partitions of the integer $\ell$, where parts represent the degrees of irreducible polynomials in the characteristic polynomial of the matrix.

3. Key Contributions and Results

A. General Linear Group and Matrix Algebra ($GL_\ell(q)$ and $M_\ell(q)$)

The paper proves that the limiting proportion of periodic points is the same for the algebra and the group.
Theorem 1: For $\ell \ge 2$,
$$\lim_{\substack{q \to \infty \\ v_L(q-1)=c}} \frac{|\text{Per}(x^L, M_\ell(q))|}{|M_\ell(q)|} = \lim_{\substack{q \to \infty \\ v_L(q-1)=c}} \frac{|\text{Per}(x^L, GL_\ell(q))|}{|GL_\ell(q)|} = \sum_{\lambda \vdash \ell} \prod_{i=1}^r \frac{1}{(\lambda_i)^{m_i} m_i!} \frac{1}{L^{m_i} v_L(q^{\lambda_i}-1)}$$
where the sum runs over all partitions $\lambda = (\lambda_1^{m_1}, \dots, \lambda_r^{m_r})$ of $\ell$.

B. Symplectic Group ($Sp_{2\ell}(q)$)

Theorem 2: The limit for the symplectic group involves a sum over partitions that account for both self-reciprocal polynomials (degree $2d_i$) and pairs of non-self-reciprocal polynomials.
$$\lim_{\substack{q \to \infty \\ v_L(q-1)=c}} \frac{|\text{Per}(x^L, Sp_{2\ell}(q))|}{|Sp_{2\ell}(q)|} = \sum_{\lambda \vdash \ell} \left( \prod_{i=1}^r \frac{1}{(\lambda_i)^{m_i} m_i!} \frac{1}{L^{m_i} v_L(q^{\lambda_i}+1)} \right) \left( \prod_{a=1}^s \frac{1}{(\lambda'_a)^{m'_a} m'_a!} \frac{1}{L^{m'_a} v_L(q^{\lambda'_a}-1)} \right)$$
Here, the partition $\lambda$ is split into parts corresponding to self-reciprocal polynomials (denoted by $\lambda_i$) and non-self-reciprocal pairs (denoted by $\lambda'_a$).

C. Unitary Group ($U_\ell(q)$)

Theorem 3: Surprisingly, the limiting value for the Unitary group is identical to that of the Symplectic group, despite the different underlying field structures ($\mathbb{F}_{q^2}$ vs $\mathbb{F}_q$) and conjugacy class definitions.
$$\lim_{\substack{q \to \infty \\ v_L(q-1)=c}} \frac{|\text{Per}(x^L, U_\ell(q))|}{|U_\ell(q)|} = \text{Same expression as Theorem 2}$$

4. Significance and Insights

1. Role of Regular Semisimple Elements: The most significant structural insight is that in the limit $q \to \infty$, only regular semisimple elements contribute to the proportion of periodic points. Matrices with nilpotent parts or zero eigenvalues become asymptotically negligible. This simplifies the complex problem of counting periodic points in non-commutative algebras to a combinatorial problem involving irreducible polynomials.

2. Universality of Limits: The paper demonstrates that the limiting proportion depends heavily on the arithmetic properties of $q$ (specifically $v_L(q-1)$) and the partition structure of the dimension, rather than the specific group type (once the group is fixed).

3. Connection to Word Maps: The work is motivated by the study of word maps on groups (e.g., commutators). Understanding the distribution of periodic points for power maps provides a foundational step for analyzing the surjectivity and image size of more complex word maps on finite groups of Lie type.

4. Methodological Rigor: The paper provides a rigorous framework for handling the asymptotic behavior of conjugacy class sizes in finite classical groups, utilizing detailed estimates of polynomial degrees and $L$-adic valuations to prove that non-semisimple contributions vanish in the limit.

In summary, Panja provides a complete asymptotic formula for the density of periodic points under power maps in major finite matrix groups and algebras, revealing a deep connection between the arithmetic of finite fields, the combinatorics of partitions, and the structure of regular semisimple elements.