[Paper Review] Linear Fractional Recurrences: Periodicities and Integrability

Eric Bedford, Kyounghee Kim|ArXiv.org|Oct 22, 2009

Fractional Differential Equations Solutions9 references3 citations

TL;DR

This paper investigates periodicity and integrability of k-step linear fractional recurrences over complex numbers using birational dynamics and cohomological degree growth. It proves that for each k, there exist k distinct recurrences with period 4k by analyzing the pullback action on Picard groups and constructing regularized models via blow-ups, establishing a link between periodicity and eigenvalue growth in the cohomology of resolved spaces.

ABSTRACT

We consider k-step recurrences of the form $z_{n+k} = A(z)/B(z)$, where A and B are linear functions of $z_n, z_{n+1}, ..., z_{n+k-1}$, which we call k-step linear fractional recurrences. The first Theorem in this paper shows that for each k there are k-step linear fractional recurrences which are periodic of period 4k. Among this class of recurrences, there is also the so-called Lyness process, which has the form $A(z)/B(z) = (a +z_{n+1} + z_{n+2} + ... + z_{n+k-1})/z_n$. The second Theorem shows that the Lyness process has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals.

Motivation & Objective

• To determine all parameter sets (α, β) for which k-step linear fractional recurrences are periodic for all initial conditions over the complex numbers.
• To extend prior results on periodicity in the k=2 and k=3 cases to general k, identifying new periodic behaviors.
• To establish a connection between periodicity and the spectral properties of the pullback map f* on Pic(X), particularly via degree growth.
• To analyze integrability of the Lyness map (h) by studying degree growth and invariant rational functions.
• To provide a systematic method for constructing invariant rational functions via homogeneous polynomials satisfying p∘f = J·p, where J is the Jacobian.

Proposed method

• Represent the recurrence as a birational map fα,β on P^k, defined by shifting indices and applying a rational function with numerator and denominator of degree k.
• Use cohomological degree growth δ(f) = lim ||(f^n)*||^{1/n} as a measure of dynamical complexity, computed via the spectral radius of f* on Pic(X).
• Construct a regularized model X via iterated blow-ups to resolve indeterminacies and ensure (f^n)* = (f*)^n on Pic(X), enabling spectral analysis.
• Identify exceptional hypersurfaces E where f(E) or f^{-1}(E) has codimension >1, and use their orbit structure to guide the blow-up process.
• For the Lyness map h, analyze degree growth of h^n by studying the action of the pullback operator T on products of linear forms.
• Construct invariant rational functions as quotients of homogeneous polynomials p satisfying p∘f = J·p, where J is the Jacobian of f.

Experimental results

Research questions

• RQ1 For which (α, β) is the k-step linear fractional recurrence (0.1) periodic for all initial conditions over C?
• RQ2 What is the maximal possible nontrivial period for such recurrences, and how does it depend on k?
• RQ3 How does the degree growth of iterates f^n relate to integrability and the existence of invariant rational functions?
• RQ4 Can the Lyness map h be shown to have quadratic degree growth for k>3 or a≠1, and what does this imply about its dynamical behavior?
• RQ5 What is the structure of rational invariants under f, and how can they be systematically constructed from solutions to p∘f = J·p?

Key findings

• For each k ≥ 1, there exist k distinct recurrences of the form (0.1) with α, β satisfying (5.3) that are periodic with period 4k.
• The degree growth δ(f) = Δ_k > 1 for generic (α, β), implying that generic recurrences are not periodic.
• The Lyness map h has quadratic degree growth in n for k > 3 or a ≠ 1, indicating non-integrability in the sense of preserving an elliptic fibration.
• For k ≥ 3, the polynomial p_0 = ∏_{j=0}^k x_j satisfies p_0∘f = J·p_0, where J is the Jacobian, and thus gives rise to an invariant rational function.
• For k ≥ 5 odd, the sum Φ_even + Φ_odd of products of linear forms satisfies the invariance condition p∘f = J·p, yielding additional invariants.
• For k > 5 even, a combination of Ψ_a and Ψ_b forms a solution to p∘f = J·p, constructed via iterated pullback T, confirming the existence of invariant rational functions.