[Paper Review] Automorphisms of rational manifolds of positive entropy with Siegel disks

Keiji Oguiso, Fabio Perroni|ArXiv.org|May 26, 2009

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper constructs higher-dimensional complex projective rational manifolds equipped with automorphisms of positive entropy that possess arbitrarily many Siegel disks—extending McMullen’s 2D examples. Using product constructions of McMullen’s rational surfaces and toric manifolds, along with number-theoretic conditions on eigenvalues, the authors realize automorphisms with exactly one or up to N Siegel disks, all arithmetic, in dimensions ≥3 and even ≥4.

ABSTRACT

Using McMullen's rational surface automorphisms, we construct projective rational manifolds of higher dimension admitting automorphisms of positive entropy with arbitrarily high number of Siegel disks and those with exactly one Siegel disk.

Motivation & Objective

• To extend McMullen’s 2-dimensional rational surface automorphisms with positive entropy and one arithmetic Siegel disk to higher dimensions.
• To investigate whether automorphisms of positive entropy on projective manifolds can possess more than one Siegel disk.
• To construct examples in dimension ≥3 with exactly one or arbitrarily many Siegel disks, all arithmetic, to resolve open questions about dynamical complexity in higher dimensions.
• To establish that positive entropy and Siegel disk coexistence is possible in higher-dimensional rational manifolds, contrasting with the zero-entropy case.

Proposed method

• Use product constructions of McMullen’s rational surface automorphisms and projective toric manifolds to build higher-dimensional manifolds.
• Apply the Künneth decomposition to compute cohomological entropy and ensure positivity via spectral radius of the induced action on cohomology.
• Employ number-theoretic conditions on eigenvalues: ensure that the product action’s eigenvalues at fixed points are multiplicatively independent algebraic integers on the unit circle.
• Construct automorphisms via compatible actions on toric varieties with local coordinates where the linearized action has eigenvalues forming a multiplicative independent unit (MAU).
• Use p-adic analysis and fixed point counting over finite fields to verify the number of fixed points and ensure Siegel disk uniqueness or multiplicity.
• Leverage the Salem trace polynomial and reciprocal polynomials to control the dynamical degree and entropy via algebraic integers.

Experimental results

Research questions

• RQ1 Can automorphisms of projective manifolds of positive entropy in dimension ≥3 have more than one Siegel disk?
• RQ2 Is it possible to construct such automorphisms with exactly one Siegel disk in even dimensions ≥4?
• RQ3 Can the number of Siegel disks in such automorphisms be made arbitrarily large?
• RQ4 What number-theoretic conditions on eigenvalues ensure the existence of multiple Siegel disks in higher-dimensional dynamical systems?
• RQ5 How does the product construction of automorphisms preserve or generate Siegel disk structure across dimensions?

Key findings

• For any integer n ≥ 4 and any N ≥ 1, there exists a non-singular complex projective rational n-fold X and an automorphism g ∈ Aut(X) with positive entropy and at least N arithmetic Siegel disks.
• There exists a 3-dimensional non-singular complex projective rational 3-fold X and an automorphism g ∈ Aut(X) with positive entropy and exactly two arithmetic Siegel disks.
• For every even integer n ≥ 4, there exists a non-singular complex projective rational n-fold X and an automorphism g ∈ Aut(X) with positive entropy and exactly one arithmetic Siegel disk.
• The construction relies on product systems of McMullen’s rational surface automorphisms and toric manifolds, with eigenvalue conditions ensuring multiplicative independence.
• The entropy h(g) is positive and equal to the logarithm of the spectral radius of g* on the total even-degree cohomology, which is an algebraic integer.
• The existence of multiple Siegel disks is guaranteed by the construction of a multiplicative independent unit (MAU) of length d+2 in the product action, ensuring local linearization with independent eigenvalues on the unit circle.