The Chebyshev polynomials $T_d$ in one variable are typical chaotic maps on $\mathbb{C}$. Chebyshev endomorphisms $P_{A_n}^d:{\mathbb{C}}^n\to {\mathbb{C}}^n$ are also chaotic. We consider the action of the dihedral group $D_{n+1}$ on ${\mathbb{C}}^n$. The endomorphism $P_{A_n}^d$ maps any $D_{n+1}$-orbit of $\mathbf{z}\in {\mathbb{C}}^n$ to a $D_{n+1}$-orbit of $P_{A_n}^d(\mathbf{z})$. The endomorphism $P_{A_n}^d$ induces a mapping on ${\mathbb{C}}^n∕D_{n+1}$.

Using invariant theory, we embed ${\mathbb{C}}^n∕D_{n+1}$ as an affine subvariety X in ${\mathbb{C}}^m$. Then we have morphisms $g_d$ on X. We study the cases $n=2$ and 3. In these cases, the morphisms $g_d$ are defined over $\mathbb{Z}$. We find a class of affine subvarieties V of X which are invariant under $g_d$. These varieties are concerned with branch loci or critical loci. The class contains ${\mathbb{C}}^2$, a cuspidal cubic, a parabola, a quadric hypersurface in ${\mathbb{C}}^4$, an affine algebraic surface in ${\mathbb{C}}^4$ which is birationally equivalent to an affine quadric cone in ${\mathbb{C}}^3$, and others. For each affine variety V in the class, there exists a polynomial parameterization $P_V$ satisfying $g_d{|}_V(P_V(y_1,\dots ,y_k))=P_V(T_d(y_1),\dots ,T_d(y_k))$, where $T_d(z)$ is a Chebyshev polynomial in one variable. Then we determine the set of bounded orbits of $g_d{|}_V$ in each invariant set V and give relations between them.