The Chebyshev polynomials in one variable are typical chaotic maps on . Chebyshev endomorphisms are also chaotic. We consider the action of the dihedral group on . The endomorphism maps any -orbit of to a -orbit of . The endomorphism induces a mapping on .
Using invariant theory, we embed as an affine subvariety X in . Then we have morphisms on X. We study the cases and 3. In these cases, the morphisms are defined over . We find a class of affine subvarieties V of X which are invariant under . These varieties are concerned with branch loci or critical loci. The class contains , a cuspidal cubic, a parabola, a quadric hypersurface in , an affine algebraic surface in which is birationally equivalent to an affine quadric cone in , and others. For each affine variety V in the class, there exists a polynomial parameterization satisfying , where is a Chebyshev polynomial in one variable. Then we determine the set of bounded orbits of in each invariant set V and give relations between them.