Dynamics of birational plane mappings. The Arnold complexity difference equation
Abstract
We consider a dynamics of a generic birational plane map n: CP2 CP2, CP2 -image of the birational mapping (inverse map is also rational)Fn : C2 C2 and its such important characteristic as the Arnold complexity CA(k), which is proportional d(k)=deg(nk)- a degree of k-iteration of the map 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 σ1 nonnegative integers m1,..., mσ1 that are determined by relations: n-mi(Oαi)=O(-1)βi, i∈(1,2,...,σ1), where n-mi is mi-iteration of inverse map, Oαi, O(-1)βi, αi, βi ∈ (1,2,...,σ), are indeterminacy points of the direct and inverse maps, σ1≤σ and σ is a number of indeterminacy points of n,n-1. If σ1 is equal to zero that d(k)=nk, 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 CP2 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.