Dynamics of split polynomial maps: uniform bounds for periods and applications

TL;DR

This paper establishes uniform bounds on the periods of certain invariant subvarieties under polynomial maps in algebraically closed fields, with applications to the dynamical Mordell-Lang problem, using classification and polynomial decomposition theories.

Contribution

It proves an effective uniform bound on periods of invariant subvarieties for disintegrated polynomial maps, extending previous dynamical systems results.

Findings

Existence of an effectively computable constant c(m,n).

Bound on periods of invariant subvarieties depending only on degrees m and n.

Application to a case of the dynamical Mordell-Lang problem.

Abstract

Let K be an algebraically closed field of characteristic 0. Following Medvedev-Scanlon, a polynomial of degree d > 1 is said to be disintegrated if neither f nor -f is linearly conjugate to x^d or T_d(x) where T_d is the Chebyshev polynomial of degree d. Let m and n be integers greater than 1, we prove that there exists an effectively computable constant c(m,n) depending only on m and n such that the following holds. Let f_1,...,f_n be polynomials with coefficients in K, which are disintegrated polynomials of degree at most m and let F be the induced coordinate-wise self-map of the n-th dimensional affine space, i.e. F(x_1,..,x_n)=(f_1(x_1),...,f_n(x_n)). Then the period of every irreducible F-periodic subvariety of the n-th dimensional affine space with non-constant projection to each coordinate axis is at most c(m,n). As an immediate application, we prove an instance of the dynamical Mordell-Lang problem following recent work of Xie. The main technical ingredients are Medvedev-Scanlon classification of invariant subvarieties together with classical and more recent results in Ritt's theory of polynomial decomposition.