Algebraic Dynamics of Polynomial Automorphisms

Algebraic dynamics of polynomial automorphisms examines the long-term behaviour of invertible polynomial self-maps on affine or projective varieties, with a focus on complex surfaces and higher-dimensional analogues. Central to this field is the study of dynamical degrees, which quantify the exponential growth of pullbacks of cohomology classes, and entropy, which measures orbit complexity. Polynomial automorphisms such as Hénon maps on C² serve as fundamental examples, exhibiting rich structures of Julia and Fatou sets, invariant currents and measures, and intricate hyperbolic dynamics. Research has advanced understanding of equidistribution of periodic points, arithmetic and geometric growth rates, and the interplay between algebraic geometry and ergodic theory. Recent work has also explored the arithmetic aspect through conjectures relating arithmetic and dynamical degrees, the classification of invariant subvarieties, and the impact of automorphism groups on manifold geometry. Overall, the field integrates techniques from complex analysis, algebraic geometry and dynamical systems to characterise stability, rigidity and bifurcation phenomena in polynomial automorphisms, with implications for number theory, moduli problems and computational dynamics.

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Algebraic Dynamics of Polynomial Automorphisms publication trend

The graph below shows the total number of articles in algebraic dynamics of polynomial automorphisms across all publications each year (not limited to Nature Index journals).

Technical terms

Polynomial automorphism: A bijective polynomial map of affine space with a polynomial inverse.

Dynamical degree: The exponential growth rate of iterates acting on cohomology or degree of pullbacks.

Hyperbolicity: A property where tangent space splits into uniformly expanding and contracting directions.

Fatou component: A maximal region in which iterates of a holomorphic map form a normal (equicontinuous) family.

Composition operator: An operator on a function space defined by pre-composition with a given self-map.

References

1. Kawaguchi–Silverman conjecture for certain surjective endomorphisms. Documenta Mathematica (2022).
2. Bounded composition operators on functional quasi-Banach spaces and stability of dynamical systems. Advances in Mathematics (2023).
3. Escaping Fatou components with disjoint hyperbolic limit sets. Mathematische Zeitschrift (2024).
4. Topological and geometric hyperbolicity criteria for polynomial automorphisms of. Ergodic Theory and Dynamical Systems (2021).