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Independence of multipliers in several variables complex dynamics

Igors Gorbovickis, Johan Taflin

TL;DR

The paper addresses the problem of independence of multipliers (periodic point eigenvalues) for regular polynomial endomorphisms of $\mathbb{C}^n$ and endomorphisms of $\mathbb{P}^n$, proving that $nN_{d,n}$ eigenvalue functions for cycles of period at least $4$ are algebraically independent; this yields local coordinates on a Zariski-open subset of the moduli space. The authors establish irreducibility of the spaces of maps with $N$ marked periodic points by proving transitivity of the monodromy on fibers, including an extension to projective space and to eigendirections. They compute and differentiate the eigenvalue maps, showing that one can choose periodic points so that these maps are locally independent, and extend the argument to conclude results for the polynomial and projective settings. These independence results feed into consequences for the bifurcation measure and the critical height, proving that the bifurcation locus has nonempty interior in higher dimensions and providing uniform control on critical preperiodic points. The work thus advances the understanding of moduli spaces and dynamical bifurcations in several complex variables, with implications for arithmetic dynamics and the geometry of parameter spaces. The results generalize prior one- and two-variable findings to $n\ge3$, enabling new corollaries about interior bifurcation regions and uniform heights in high dimension.

Abstract

We establish the independence of multipliers for polynomial endomorphisms of $\mathbb C^n$ and endomorphisms of $\mathbb P^n.$ This allows us to extend results about the bifurcation measure and the critical height obtained in \cite{arXiv:2305.02246} to the case of polynomial endomorphisms of $\mathbb C^n$ for $n\geq 3$. An important step in the proof is the irreducibility of the spaces of endomorphisms with $N$ marked periodic points, which is of independent interest.

Independence of multipliers in several variables complex dynamics

TL;DR

The paper addresses the problem of independence of multipliers (periodic point eigenvalues) for regular polynomial endomorphisms of and endomorphisms of , proving that eigenvalue functions for cycles of period at least are algebraically independent; this yields local coordinates on a Zariski-open subset of the moduli space. The authors establish irreducibility of the spaces of maps with marked periodic points by proving transitivity of the monodromy on fibers, including an extension to projective space and to eigendirections. They compute and differentiate the eigenvalue maps, showing that one can choose periodic points so that these maps are locally independent, and extend the argument to conclude results for the polynomial and projective settings. These independence results feed into consequences for the bifurcation measure and the critical height, proving that the bifurcation locus has nonempty interior in higher dimensions and providing uniform control on critical preperiodic points. The work thus advances the understanding of moduli spaces and dynamical bifurcations in several complex variables, with implications for arithmetic dynamics and the geometry of parameter spaces. The results generalize prior one- and two-variable findings to , enabling new corollaries about interior bifurcation regions and uniform heights in high dimension.

Abstract

We establish the independence of multipliers for polynomial endomorphisms of and endomorphisms of This allows us to extend results about the bifurcation measure and the critical height obtained in \cite{arXiv:2305.02246} to the case of polynomial endomorphisms of for . An important step in the proof is the irreducibility of the spaces of endomorphisms with marked periodic points, which is of independent interest.

Paper Structure

This paper contains 21 sections, 26 theorems, 127 equations.

Table of Contents

  1. Introduction, statement of the main results
  2. Independence of the eigenvalues
  3. Previous results and consequences
  4. Irreducibility of varieties of marked periodic orbits over $\mathcal{P}_d^n$ and $\mathrm{End}_d(\mathbb{P}^n)$
  5. Strategy of the proof of Theorem \ref{['main_theorem_1']} and structure of the paper
  6. Irreducibility of $\mathcal{P}_d^n$ with marked periodic points
  7. Strategy of the proof of Theorem \ref{['main_theorem_2']}
  8. Hyperbolicity condition for compositions of polynomials
  9. Main steps of the proof: construction of the loops
  10. Extension of Theorem \ref{['main_theorem_2']} to the space $\mathrm{End}_d(\mathbb{P}^n)$
  11. Proof of Theorem \ref{['main_theorem_3']}
  12. The diagonal entries of the Jacobian and the eigenvalue maps
  13. Partial derivatives of the maps $\rho_{k,\mathbf{w}_0}$
  14. Independence of the maps $\rho_{k,\mathbf{w}_0}$ for selected periodic orbits
  15. Proof of Theorem \ref{['main_theorem_1']} and Theorem \ref{['main_theorem_Pn']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $d$ and $n$ be any pair of integers, such that $d\ge 2$ and $n\ge 1$. Then any $nN_{d,n}$ distinct eigenvalue functions defined on $\mathcal{P}_d^n$ and corresponding to distinct cycles of periods not smaller than $4$, are algebraically independent over $\mathbb{C}$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Theorem 2.1
  • Lemma 2.2
  • ...and 45 more