No Smooth Julia Sets for Polynomial Diffeomorphisms of ${\mathbb C}^2$ with Positive Entropy

Abstract

For any polynomial diffeomorphism f of ${\mathbb C}^2$ with positive entropy, the Julia set of f is never $C^1$ smooth as a manifold-with-boundary.

Appendix: Nonsmoothness of J, \(J^*\), and K

Let us turn our attention to other dynamical sets for polynomial diffeomorphisms of positive entropy. These are $J:=J^+\cap J^-$, $K:= K^+\cap K^-$, and the set $J^*$, which coincides with the closure of the set of periodic points of saddle type. (See [3, 5], and [2] for other characterizations of $J^*$.) We have $J^*\subset J\subset K$. We note that none of these sets can be a smooth 3-manifold: otherwise, for any saddle point p, it would be a bounded set containing $W^s(p)$ or $W^u(p)$, which is the holomorphic image of ${\mathbb C}$. The following was suggested by Remark 5.9 of Cantat in [9]; we sketch his proof:

Proposition 6.1

If $J=J^*$, then it is not a smooth 2-manifold.

Proof

Let p be a saddle point, and let $W^u(p)$ be the unstable manifold. The slice $J \cap W^u(p)$ is smooth and invariant under multiplication by the multiplier of Df. This means that in fact, the multiplier must be real, and the restriction of $G^+$ to the slice must be linear on each (half-space) component of $W^u(p)- J$.

The identity $G^+\circ f = d\cdot G^+$ means that the canonical metric (defined in [6]) is multiplied by d. Thus f is quasi-expanding on $J^*$. Now, applying this argument to $f^{-1}$ we get that f is quasi-hyperbolic. Further, $J^*=J$, so it is quasi-hyperbolic on J. If f fails to be hyperbolic, then by [7] there will be a one-sided saddle point, which cannot happen since J is smooth.

Now that f is hyperbolic on J, there is a splitting $E^s \oplus E^u$ of the tangent bundle, so we conclude that J is a 2-torus. The dynamical degree must be the spectral radius of an invertible 2-by-2 integer matrix, but this means it is not an integer, which contradicts the fact the dynamical degree of a Hénon map is its algebraic degree.

Proposition 6.2

Suppose that the complex Jacobian is not equal to $\pm 1$. Then for each saddle (periodic) point p and each neighborhood U of p, neither $J\cap U$ nor $J^*\cap U$ nor $K\cap U$ is a $C^1$ smooth 2-manifold.

Proof

Let us write $M:=J\cap U$ and $g:=f|_M$. (The following argument works, too, if we take $M=J^*\cap U$ or $M=K\cap U$.) The tangent space $T_pM$ is invariant under Df. The stable/unstable spaces $E^{s/u}\subset T_p{\mathbb C}^2$ are invariant under $D_pf$. The space $E^s$ (or $E^u$) cannot coincide with $T_pM$, for otherwise the complex stable manifold $W^s(p)$ (or $W^u(p)$) would be locally contained in M, and thus globally contained in J. But the $W^{s/u}$ are uniformized by ${\mathbb C}$, whereas J is bounded. We conclude that p is a saddle point for g, and thus the local stable manifold $W^s_{\mathrm{loc}}(p; g)$ is a $C^1$-curve inside the complex stable manifold $W^s(p)$. As in Lemma 4.3, we conclude that the multiplier for $D_pf|_{E^u_p}$ is $\pm d$ and the multiplier for $D_pf|_{E^s_p}$ is $\pm 1/d$. Thus the complex Jacobian is $\delta =\pm 1$.

Solenoids The two results above concern smoothness, but no example is known where J, $J^*$, or K is even a topological 2-manifold. In the cases where $J^+$ has been shown to be a topological 3-manifold (see [8, 11, 16, 20]) it also happens that J is a (topological) real solenoid, and in these cases it is also the case that $J=J^*$. Further, for every saddle (periodic) point p, there is a real arc $\gamma _p=W^u_{\mathrm{loc}}(p)\cap J$. If we apply the argument of Proposition 6.2 to this case, we conclude that $\gamma _p$ is not $C^1$ smooth.