Some rigidity results for polynomial automorphisms of $\mathbf{C}^2$
Abstract
We prove several new rigidity results for polynomial automorphisms of $\mathbf{C}^2$ with positive entropy. A first result is that a complex slice of the (forward or backward) Julia set is never a smooth, or even rectifiable, curve. We also show that such an automorphism cannot preserve a global holomorphic foliation, nor a real-analytic foliation with complex leaves. These results are used to show that under mild assumptions, two real-analytically conjugate automorphisms are polynomially conjugate. The properties of multipliers of saddle periodic orbits play an important role in these results. We also study some arithmetic questions related to these multipliers.
Fields of Study & Subjects
Advanced Differential Equations and Dynamical Systems
Quantum chaos and dynamical systems
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- Cambridge Journal of Mathematics
- Volume
- 14
- Issue
- 3
- Pages
- 539–601
- Publication date
- 1 January 2026
- Language
- EN