Dynamics of birational plane mappings. The Arnold complexity difference equation

TL;DR

This paper investigates the growth of Arnold complexity in birational plane maps, deriving a linear difference equation and linking the growth rate to algebraic-geometric properties and Veselov's conjecture.

Contribution

It formulates an autonomous linear difference equation for Arnold complexity growth based on indeterminacy points, connecting algebraic geometry with dynamical systems theory.

Findings

Derived a linear difference equation for d(k)

Connected growth of d(k) to root spectrum of the secular equation

Linked Veselov's conjecture to root spectrum properties

Abstract

We consider a dynamics of a generic birational plane map \Phi_n: CP^2 \to CP^2, CP^2 -image of the birational mapping (inverse map is also rational)F_n : C^2 \to C^2 and its such important characteristic as the Arnold complexity C_A(k), which is proportional d(k)=deg(\Phi_n^k)- a degree of k-iteration of the map \Phi_n, on the basis on algebraic-geometrical properties of such maps. Additional importance of this characteristic follows from the Veselov conjecture about the polynomial boundedness of the growth of d(k) for integrable dynamical systems with a discrete time defined by birational plane maps. The autonomous linear difference equation with integer coefficients for d(k) is obtained. This equation is fully defined by \sigma_1 nonnegative integers m_1,..., m_\sigma_1 that are determined by relations: \Phi_n^{-m_i}(O_\alpha_i)=O^(-1)_\beta_i, i\in(1,2,...,\sigma_1), where \Phi_n^{-m_i} is m_i-iteration of inverse map, O_\alpha_i, O^(-1)_\beta_i, \alpha_i, \beta_i \in (1,2,...,\sigma), are indeterminacy points of the direct and inverse maps, \sigma_1\leq\sigma and \sigma is a number of indeterminacy points of \Phi_n,\Phi_n^{-1}. If \sigma_1 is equal to zero that d(k)=n^k, otherwise the growth of d(k) is fully defined by a root spectrum of the secular equation associated with the difference equation for d(k). The Veselov conjecture corresponds to the root spectrum consisting of values being equal to modulo one. The author doesn't suppose that the reader has acquaintance with the algebraic geometry (AG) in CP^2 and the dynamical systems theory (DST) or the functional equations since in the paper there are given all needed definitions of used concepts of AG or DST and theorems.