On the dynamics of real quadratic harmonic polynomials with tree Julia sets

Abstract Polynomial Julia sets with tree structure, typically Hubbard trees, play an important role in holomorphic dynamics. In this paper, we study the dynamics of ${f_\alpha (z)=z^2+\alpha \bar {z}}$ for $\alpha $ being real and the Julia sets being trees. We show that all such $\alpha $ form an interval $[1,4]$ . This answers a question of G. Sienra. We further show that $f_\alpha $ exhibits non-trivial dynamics on the Fatou set which equals to the escaping set.