[Paper Review] Good Reduction of Periodic Points

Benjamin Hutz|arXiv (Cornell University)|Jan 23, 2008

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper establishes bounds on the primitive period of periodic points under iteration of a morphism on a projective variety over a local field, generalizing classical results from rational maps on P¹. It shows that the period over the base field is either equal to or a multiple of the period over the residue field, with the multiplier determined by the action on the cotangent space and the residue characteristic, leading to effective uniform bounds on periodic points under good reduction.

ABSTRACT

We consider the dynamical system created by iterating a morphism of a projective variety defined over the field of fractions of a discrete valuation ring. We study the primitive period of a periodic point in this field in relation to the primitive period of the reduced point in the residue field, the order of the action on the cotangent space, and the characteristic of the residue field.

Motivation & Objective

• To extend known bounds on periodic points in dynamical systems from P¹ to higher-dimensional projective varieties.
• To understand how the primitive period of a K-rational periodic point relates to its reduction modulo the maximal ideal of a discrete valuation ring.
• To establish effective upper bounds on the primitive period of periodic points under good reduction, generalizing results from rational maps to higher-dimensional varieties.
• To analyze the role of the cotangent space action and residue characteristic in period growth during reduction.
• To prove that for étale morphisms, the period is bounded even when the variety is not necessarily smooth or irreducible.

Proposed method

• Define good reduction for a projective variety X/K and a morphism φ:X→X using smooth proper models over the ring R of integers of a local field K.
• Use the reduction map to relate periodic points in X(K) to their images in the special fiber X×k, analyzing the primitive period m of the reduced point.
• Analyze the action of the iterate φ^m on the cotangent space at the orbit of a periodic point, identifying a φ-stable subspace V with order r_V.
• Derive recursive inequalities for valuation growth in iterates using the p-adic valuation of the derivative and binomial coefficients.
• Apply Fibonacci-like recurrence relations for p=2 and exponential growth for p≠2 to bound the exponent e in the period formula n = m r_V p^e.
• Use the structure of the orbit scheme and its special fiber to relate the dimension d′ of the cotangent subspace V to the upper bound (Nπ)^{d′}−1.

Experimental results

Research questions

• RQ1 How does the primitive period of a K-rational periodic point relate to the primitive period of its reduction modulo the maximal ideal of a discrete valuation ring?
• RQ2 What role does the action of the morphism on the cotangent space of the orbit play in determining the period growth under reduction?
• RQ3 Can effective upper bounds be established for the primitive period of periodic points on higher-dimensional projective varieties under good reduction?
• RQ4 How does the residue characteristic p influence the possible period growth, particularly in the case of p=2?
• RQ5 To what extent can the bounds be extended to non-smooth or reducible varieties, especially under étale morphisms?

Key findings

• The primitive period n of a K-rational periodic point satisfies n = m r_V p^e, where m is the reduced period, r_V is the order of the induced map on a φ-stable subspace of the cotangent space, and e ≥ 0.
• For p ≠ 2, the exponent e satisfies e ≤ 1 + log₂(v(p)); for p = 2, e ≤ 1 + log_α( (sqrt(5)v(2) + sqrt(5v(2)² + 4))/2 ), where α = (1+√5)/2.
• When working over Q, the exponent e is bounded by 1 for p ≠ 2 and by 3 for p = 2.
• The period n is bounded by n ≤ p^{d+1}(p^d − 1) for p ≠ 2 and n ≤ 2^{d+3}(2^d − 1) for p = 2, where d is the dimension of the variety.
• For étale morphisms, the period is bounded even when the variety is not smooth or irreducible, provided the morphism stabilizes a smooth irreducible component with good reduction.
• The bound on the number of irreducible components of a variety defined by degree-d forms (≤ d^N) implies that the number of possible periods is finite under good reduction.