Deautonomisation by singularity confinement and degree growth

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Summary

The paper discusses the relationship between singularity confinement and degree growth in birational mappings of the plane. It provides a rigorous formulation and explanation of the mechanism by which the dynamical degree is reflected in confinement conditions. The authors introduce the concept of a space of initial conditions for a non-autonomous mapping and show that it is equivalent to the existence of a rational surface on which the mapping becomes an automorphism.

Highlights

• The paper provides a rigorous formulation and explanation of the mechanism by which the dynamical degree is reflected in confinement conditions.
• The authors introduce the concept of a space of initial conditions for a non-autonomous mapping.
• The space of initial conditions is equivalent to the existence of a rational surface on which the mapping becomes an automorphism.
• The authors show that the dynamical degree of the mapping is given by the largest eigenvalue of the action of the mapping on the Picard lattice.
• The paper provides a classification of anticanonical divisors for spaces of initial conditions that correspond to non-integrable mappings.
• The authors introduce the concept of a sufficient subset of Q⊥ and show that it is sufficient to detect the dynamical degree.
• The paper provides examples of mappings of QRT Class III and Class I form to demonstrate the results.

Key Insights

• The concept of a space of initial conditions is crucial in understanding the relationship between singularity confinement and degree growth in birational mappings of the plane.
• The dynamical degree of the mapping is given by the largest eigenvalue of the action of the mapping on the Picard lattice, which provides a way to compute the degree growth of the mapping.
• The classification of anticanonical divisors for spaces of initial conditions that correspond to non-integrable mappings provides a way to understand the geometry of the rational surfaces associated with these mappings.
• The concept of a sufficient subset of Q⊥ provides a way to detect the dynamical degree of the mapping, which is a key insight in understanding the relationship between singularity confinement and degree growth.
• The examples of mappings of QRT Class III and Class I form demonstrate the results of the paper and provide a way to understand the relationship between singularity confinement and degree growth in specific cases.
• The paper provides a rigorous formulation and explanation of the mechanism by which the dynamical degree is reflected in confinement conditions, which is a key insight in understanding the relationship between singularity confinement and degree growth.
• The authors' use of algebraic geometry and lattice theory provides a powerful tool for understanding the geometry of the rational surfaces associated with birational mappings of the plane.

Mindmap

Citation

Stokes, A., Mase, T., Willox, R., & Grammaticos, B. (2023). Deautonomisation by singularity confinement and degree growth (Version 3). arXiv. https://doi.org/10.48550/ARXIV.2306.01372