Dynamics of Chebyshev endomorphisms on some affine algebraic varieties

Abstract

The Chebyshev polynomials Td in one variable are typical chaotic maps on C. Chebyshev endomorphisms PAnd: Cn→Cn are also chaotic. We consider the action of the dihedral group Dn+1 on Cn. The endomorphism PAnd maps any Dn+1-orbit of z∈Cn to a Dn+1-orbit of PAnd(z). The endomorphism PAnd induces a mapping on Cn∕Dn+1. Using invariant theory, we embed Cn∕Dn+1 as an affine subvariety X in Cm. Then we have morphisms gd on X. We study the cases n=2 and 3. In these cases, the morphisms gd are defined over Z. We find a class of affine subvarieties V of X which are invariant under gd. These varieties are concerned with branch loci or critical loci. The class contains C2, a cuspidal cubic, a parabola, a quadric hypersurface in C4, an affine algebraic surface in C4 which is birationally equivalent to an affine quadric cone in C3, and others. For each affine variety V in the class, there exists a polynomial parameterization PV satisfying gd|V(PV(y1,…,yk))=PV(Td(y1),…,Td(yk)), where Td(z) is a Chebyshev polynomial in one variable. Then we determine the set of bounded orbits of gd|V in each invariant set V and give relations between them.