Polynomial contractions of $\mathbb C^d$ and degree growth

Abstract: We give a simple example of a polynomial contraction automorphism of $\mathbb C^d$, $ d\ge 3 $, with unbounded degree growth. Combined with Poincaré-Dulac theorem it provides an algebraic automorphism of $\mathbb C^d$, $ d\ge 3 $, which is holomorphically but not algebraically linearizable.

• The paper provides explicit constructions of polynomial contraction automorphisms on C^d (d≥3) that exhibit unbounded, linear degree growth, with degree γⁿ = n+1.
• It establishes that bounded degree growth is equivalent to strict algebraicity, thereby clarifying the impact of algebraic conjugacy on polynomial automorphisms.
• The study distinguishes holomorphic linearizability from algebraic linearizability using the Poincaré–Dulac theorem and non-resonance conditions on eigenvalues.

Polynomial Contractions of $\mathbb C^d$ and Degree Growth

Introduction and Motivation

The paper addresses a question concerning polynomial contractions on complex affine spaces $\mathbb C^d$ for $d\geq 3$ and their degree growth under iteration. Specifically, it provides explicit constructions of polynomial automorphisms—contractions—exhibiting unbounded degree growth, and analyzes the linearizability properties connected to algebraic and holomorphic conjugacy. The results connect to broader themes in dynamical systems, algebraic group actions, and the GAGA-type correspondence for complex manifolds endowed with contraction automorphisms.

Strict Algebraicity and Degree Growth

The concept of strict algebraicity in polynomial automorphisms is formalized via actions by algebraic groups. The main proposition establishes an equivalence: a polynomial automorphism is strictly algebraic if and only if the sequence of degrees of its iterates is uniformly bounded. The argument leverages the Zariski closure within spaces of bounded degree automorphisms and classical results about locally finite actions of algebraic groups on coordinate rings. This invariant property further extends to algebraic conjugacy—bounded degree growth is preserved under polynomial automorphism conjugation.

Contractions in Dimension Two and Higher

For $d=2$, every contraction automorphism is either affine or elementary, with constant degree growth. Loxodromic automorphisms, which exhibit exponential degree growth and positive topological entropy, cannot be contractions due to their periodic points and entropy properties, as established via classical dynamical systems results (Milnor, Friedland, Smillie).

In contrast, for $d\geq 3$, explicit affine-triangular contractions exhibit linear degree growth. The constructed automorphism on $\mathbb C^3$ is:

$$(x, y, z) \mapsto (\lambda_1 (y + xz), \lambda_2 x, \lambda_3 z),\quad 0 < \lambda_i < \frac{1}{2}$$

Iterations yield $\deg \gamma^n = n+1$, demonstrating unbounded degree growth. The contraction property is confirmed via recursive analysis of the dynamics, showing convergence to zero. The construction generalizes to arbitrary dimensions via additional coordinate scalings.

Strong numerical results include:

• Explicit linear degree growth: $\deg \gamma^n = n+1$
• For $d=2$, impossibility of unbounded degree growth for contractions

Holomorphic Versus Algebraic Linearizability

The Poincaré–Dulac theorem is invoked to show that, under suitable choices of scaling parameters $\mathbb C^d$0 (algebraically independent, with $\mathbb C^d$1), the contraction automorphism is holomorphically but not algebraically linearizable. The non-resonance condition on the eigenvalues at the fixed point ensures holomorphic conjugacy to a linear map, yet no polynomial conjugacy exists due to unbounded degree growth.

This establishes a critical dichotomy:

• Holomorphic Linearizability: Possible via analytic transformations when the resonance relation fails among eigenvalues.
• Algebraic Linearizability: Impossible for constructed contractions, as algebraic conjugacy requires bounded degree growth, contradicting the explicit construction.

Implications and Future Perspectives

The results clarify structural differences between holomorphic and algebraic automorphism groups on affine and complex manifolds. The constructed example explicitly answers a question posed by Ornea and Verbitsky regarding GAGA-type theorems for manifolds with holomorphic contractions, showing how algebraic constraints such as degree growth impose limits on correspondences between algebraic and analytic categories.

From a dynamical perspective, understanding degree growth of automorphisms informs entropy and orbit structure analyses in multidimensional contexts. The paper’s constructions could potentially inform classification of automorphisms, characterization of ind-group actions, and contribute to the development of algebraic dynamical systems with modulated complexity.

Theoretically, future developments may explore:

• Finer properties of degree growth in higher dimensions and its relation to dynamical invariants
• Extension of holomorphic versus algebraic linearizability distinctions to other classes of automorphisms and contractions in more general complex varieties
• Consequences for topological entropy and orbit distribution in formally vs. strictly algebraic contexts

Practically, understanding these contraction dynamics could impact algorithmic approaches for automorphism detection and canonical form identification in symbolic computation environments, and potentially inform geometric and combinatorial aspects of moduli spaces with contraction group actions.

Conclusion

The paper delineates the existence of polynomial contraction automorphisms of $\mathbb C^d$2 ($\mathbb C^d$3) with unbounded degree growth, explicitly constructing examples exhibiting linear growth. It further demonstrates the fundamental distinction between holomorphic and algebraic linearizability, grounded in degree growth behavior and resonance conditions. These results have substantive implications for the interplay between analytic and algebraic automorphism structures, dynamical invariants, and algebraic group actions within higher-dimensional complex geometry.