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Functional equations in formal power series

Fedor Pakovich

TL;DR

This work develops an algebraic framework for functional equations in the semigroup $oldsymbol{ extGamma}$ of order at least two formal power series under composition. Central to the approach are Böttcher functions $eta_A$ that conjugate elements of $oldsymbol{ extGamma}$ to monomial maps, and the transition groups $G_A$ capturing internal symmetries; these tools yield precise solvability criteria and explicit solutions for equations of the form $F=A ound X$, $F=X ound A$, and $X ound A=Y ound B$. The results classify decompositions of elements of $oldsymbol{ extGamma}$ via ordered factorizations of the order, characterize symmetric (e.g., even/odd) series, and establish a complete description of right amenability and reversibility in this setting, including a Horwitz–Rubel-type outcome for decompositions in $oldsymbol{ extGamma}$. Together, these contributions provide a powerful algebraic analogue to decomposition theories in rational functions, with broader implications for dynamical systems and semigroup actions on formal power series. The framework links Böttcher conjugacy, transition structures, and symmetry constraints to deliver concrete, computable criteria and constructions across decompositions, symmetries, and amenability questions.

Abstract

Let $k$ be an algebraically closed field of characteristic zero, and $k[[z]]$ the ring of formal power series over $k$. In this paper, we study equations in the semigroup $z^2k[[z]]$ with the semigroup operation being composition. We prove a number of general results about such equations and provide some applications. In particular, we answer a question of Horwitz and Rubel about decompositions of ``even'' formal power series. We also show that every right amenable subsemigroup of $z^2k[[z]]$ is conjugate to a subsemigroup of the semigroup of monomials.

Functional equations in formal power series

TL;DR

This work develops an algebraic framework for functional equations in the semigroup of order at least two formal power series under composition. Central to the approach are Böttcher functions that conjugate elements of to monomial maps, and the transition groups capturing internal symmetries; these tools yield precise solvability criteria and explicit solutions for equations of the form , , and . The results classify decompositions of elements of via ordered factorizations of the order, characterize symmetric (e.g., even/odd) series, and establish a complete description of right amenability and reversibility in this setting, including a Horwitz–Rubel-type outcome for decompositions in . Together, these contributions provide a powerful algebraic analogue to decomposition theories in rational functions, with broader implications for dynamical systems and semigroup actions on formal power series. The framework links Böttcher conjugacy, transition structures, and symmetry constraints to deliver concrete, computable criteria and constructions across decompositions, symmetries, and amenability questions.

Abstract

Let be an algebraically closed field of characteristic zero, and the ring of formal power series over . In this paper, we study equations in the semigroup with the semigroup operation being composition. We prove a number of general results about such equations and provide some applications. In particular, we answer a question of Horwitz and Rubel about decompositions of ``even'' formal power series. We also show that every right amenable subsemigroup of is conjugate to a subsemigroup of the semigroup of monomials.
Paper Structure (13 sections, 32 theorems, 156 equations)

This paper contains 13 sections, 32 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Böttcher functions
  3. Lemmata about formal power series
  4. Böttcher functions and the equation $A\circ X=Y\circ B$
  5. Transition functions
  6. Functional equations $F=A\circ X$ and $F=X\circ A$
  7. Equivalency classes of decompositions of formal power series
  8. Formal power series with symmetries
  9. Characterizations of formal powers series with symmetries
  10. Decompositions of formal powers series with symmetries
  11. Functional equation $X\circ A=Y\circ B$ and reversibility
  12. Functional equation $X\circ A=Y\circ B$
  13. Right reversibility of subsemigroups of $\Gamma$

Key Result

Theorem 1.1

Let $A\in \Gamma$ be a formal power series of order $n$, and $\beta_A$ some Bötcher function. Then every decomposition of $A$ into a composition of elements $A_1, A_2, \dots ,A_r$ of $\Gamma$ is equivalent to the decomposition Thus, equivalence classes of decompositions of $A$ are in a one-to-one correspondence with ordered factorizations of $n$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 22 more