Dynamics of Word Maps on Groups and Polynomial Maps on Algebras

2025/11/06 by Saikat Panja, Panja, Saikat
Mathematics · #Mathematical Dynamics and Fractals #Advanced Algebra and Geometry #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2511.04377

Abstract

We introduce the notions of Fatou and Julia sets in the context of word maps on complex Lie groups and polynomial maps on finite-dimensional associative \mathbb C-algebras. For the group-theoretic question, we investigate the dynamics of the power map x ↦ x^M on the Lie group GL_n(\mathbb C), where M ≥ 2 is an integer. For the algebra-related question, we study polynomial self-maps of M_n(\mathbb C) induced by monic polynomials in one variable. In both cases, we pin down the explicit description of the Fatou and Julia sets. We also show that there does not exist any wandering Fatou component of the pair (p,\mathrm M_n(\mathbb C)) where p∈\mathbb C[z] is a monic polynomial of degree ≥ 2.

Citations

• Power maps on General Linear groups over finite principal ideal local rings of length two
• Polynomial maps with constants on split octonion algebras
• Word maps, polynomial maps and image ratios
• Fibers of the square map in some finite groups of Lie type and an application
• Matrix evaluations of noncommutative rational functions and Waring problems
• Images of polynomial maps with constants
• Surjectivity of polynomial maps on Matrices
• Powers in finite unitary groups
• Powers in finite orthogonal and symplectic groups: A generating function approach
• The Waring Problem for Matrix Algebras
• Commutators and images of noncommutative polynomials
• Fatou-Julia dichotomy of matrix-valued polynomials
• Word maps, word maps with constants and representation varieties of\n one-relator groups

Related

• Dynamics of groups of automorphisms of character varieties and Fatou/Julia decomposition for Painlevé 6
• Wandering Fatou components on p-adic polynomial dynamics
• A polynomial automorphism with a wandering Fatou component
• Geometry and arithmetic of verbal dynamical systems on simple groups
• Wandering Fatou Components and Algebraic Julia Sets