Smoothness of the Julia Set

In dimension 1, it is possible for a Julia set to be an interval (Chebyshev polynomial) or a circle ($z\mapsto z^d$). It is natural to ask how regular the sets $J^+$ and $J^-$ might be for Hénon maps. Fornaess and Sibony observed that for a generic Hénon map, the Julia sets $J^+$ and $J^-$ are not $C^1$ smooth. Of course, a Hénon map with smooth Julia set would be expected to be non-generic and special in some way. Thus we can ask whether such special maps might exist.

In the direction "yes," we may consider maps of the form $f(x,y)=(p(x)-\delta y,x)$, where $|\delta|$ will be taken sufficiently small. Further, we suppose that the Julia set $J_p\subset{\Bbb C}$ is a Jordan curve, and $p$ is uniformly expanding on $J_p$. In this case, Hubbard and Oberste-Vorth and Fornaess and Sibony showed that $J^+$ is a topological 3-manifold. In the non-hyperbolic case, Radu and Tanase have shown that $J^+$ is a topological 3-manifold for certain perturbations of $p(z)= z^2+{1\over 4}$.

However, it was shown with Kyounghee Kim that it is never the case that $J^+$ is $C^1$ smooth (as a manifold-with-boundary, in case $J^+$ would have boundary).

For endomorphisms of ${\Bbb C}^2$, it is possible for endomorphisms to have interesting semi-algebraic Julia sets. The (few) examples that are known have singularities. So it is natural to ask whether in the case of Hénon maps $J^+$ or $J^-$ might be semi-analytic. Such sets are defined by equations $\{r(z,\bar z)=0\}$, as well as inequalities $\{s(z,\bar z)\ge 0\}$, where $r(z,\bar z)$ and $s(z,\bar z)$ are locally defined by convergent power series. However, another work with Kyounghee Kim shows that the Julia set cannot be semianalytic.

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