Generalized Chebyshev maps of $\mathbf{C}^{2}$ and their perturbations

Abstract

The Chebyshev map is a typical chaotic map. We consider generalized Chebyshev maps $T_{k}$ of $\mathbf{C}^{2}$. The support of the maximal entropy measure of $T_{k}$ is connected. We perturb $T_{k}$ in a certain direction. Then we can show the support of the maximal entropy measure of this map is a Cantor set.