We explore the connected/disconnected dichotomy for the Julia set of polynomial automorphisms of C^2. We develop several aspects of the question, which was first studied by Bedford-Smillie. We introduce a new sufficient condition for the connectivity of the Julia set, that carries over for certain H{é}non-like and birational maps. We study the structure of disconnected Julia sets and the associated invariant currents. This provides a simple approach to some results of Bedford-Smillie, as well as some new corollaries --the connectedness locus is closed, construction of external rays in the general case, etc.
We also prove the following theorem: a hyperbolic polynomial diffeomorphism of C^2 with connected Julia set must have attracting or repelling orbits. This is an analogue of a well known result in one dimensional dynamics.
For any polynomial diffeomorphism $f$ of ${\Bbb C}^2$ with positive entropy, neither the Julia set of $f$ nor of its inverse $f^{-1}$ is semi-analytic.
For any polynomial diffeomorphism $f$ of $\mathbb{C}^2$ with positive entropy, neither the Julia set of $f$ nor of its inverse $f^{-1}$ is $C^1$ smooth as a manifold-with-boundary.
Let f and g two rational functions having the same Julia set J_f. Lets suppose that f has a rational indifferent periodic point and that the critical set of f is disjoint of J_f. Then or J_f has to be equal to P^1, a circle, an arc of a circle for some cordinate or f and g has to verify an equation of the type: $$f^{m_1}\circ g\circ f^{m_2}\circ\cdotc\circ f^{m_n}\circ g=f^m.$$
Given a polynomial diffeomorphism f: C^2 -> C^2 there is a set $J_f\subset{\bf C}^2$ which we call the Julia set of f. The set $J_f\subset C^2$ plays the role of the Julia set $J\subset{\bf C}$ for a polynomial map of C. In the study of polynomial maps of C a great deal of attention has been paid to the connectivity of the Julia set. The focus of this paper is to investigate the J-connected/J-disconnected dichotomy in the case of polynomial diffeomorphisms of C^2. The Jacobian determinant of f is constant. We make the standing assumption that $|det\ Df|\le 1$ (this can always be achieved by re…
In this paper we continue our study of polynomial diffeomorphisms of C^2. Let us recall that there is an invariant measure $μ$, which is the pluri-complex version of the harmonic measure of the Julia set for polynomial maps of C. In this paper we give an integral formula for the Lyapunov exponents of a polynomial automorphism with respect to $μ$ analogous to the Brolin-Manning formula polynomial maps of C. Our formula relates the Lyapunov exponents to the value of a Green function at a type of critical point which we define in this paper. We show that these critical points have a very natural …
The McMullen-Sullivan holomorphic motion for topologically conjugate, complex polynomials with connected Julia set follows level sets of the Bottcher coordinate. The Buzzard-Verma holomorphic motion for hyperbolic, unstably connected, polynomial diffeomorphisms of C^2 follows level sets of the Bedford-Smillie solenoid map. It follows that this solenoid map is a conjugacy for those Henon maps that are perturbations of (one-dimensional) hyperbolic maps with connected Julia set.
We show that a strong partially hyperbolic diffeomorphism of $\mathbb{T}^3$ isotopic to Anosov has a unique quasi-attractor. Moreover, we study the entropy of the diffeomorphism restricted to this quasi-attractor.
We show that there is no C^{k+1} diffeomorphism of the k-torus which is semiconjugate to a minimal translation and has a wandering domain all of whose iterates are Euclidean balls.
Let $M$ be a closed oriented $C^\infty$ manifold and $f$ a $C^\infty$ Anosov diffeomorphism on $M$. We show that if $M$ is the two torus $T^2$, then $f$ is conjugate to a hyperbolic automorphism of $T^2$, either by a $C^\infty$ diffeomorphism or by a singular homeomorphism. We also show that for general $M$, if $f$ admits an absolutely continuous invariant measure $μ$, then $μ$ is a $C^\infty$ volume. The proofs are concatenations of well known results in the field.
Let G a group of germs of analytic diffeomorphisms in (C^2,0). We find some remarkable properties supposing that G is finite, linearizable, abelian nilpotent and solvable. In particular, if the group is abelian and has a generic dicritic diffeomorphisms, then the group is a subgroup of a 1-parametric group. In addition, we study the topological behaviour of the orbits of a dicritic diffeomorphisms. Last, we find some invariants in order to know when two diffeomorphisms are formally conjugates.
We prove that for a polynomial diffeomorphism of C^2, uniform hyperbolicity on the set of saddle periodic points implies that saddle points are dense in the Julia set. In particular f satisfies Smale's Axiom A on C^2 .
Let $f$ be an endomorphism of $\mathbb{C}\mathbb{P}^k$ of degree $>1$, and assume that for any cyclic Fatou component $W$ of $f$ having a period $p\in\mathbb{N}$, the equilibrium measure $μ_f$ has a positive charge on the boundary of $W$ if and only if $f^{-p}(W)=W$. Then we obtain a locally uniform a priori bound of the dynamics of $f$, which in particular yields a Diophantine-type estimate of the dynamics of $f$ on its domaines singuliers. We also point out that in the case of $k=1$, the statement of our assumption is related to both the impossibility for the Julia set of $f$ to be the bound…
We prove that the Bruno-R{ü}ssmann condition is optimal for the analytic preservation of a quasi-periodic invariant curve for an analytic twist map. The proof is based on Yoccoz's corresponding result for analytic circle diffeomorphisms and the uniqueness of invariant curves with a given irrational rotation number. We also prove a similar result for analytic Tonelli Hamiltonian flow with n = 2 degrees of freedom; for n $\ge$ 3 we only obtain a weaker result which recovers and slightly improves a theorem of Bessi.
For a generic vector field robustly without horseshoes, and an aperiodic chain recurrent class with singularities whose saddle values have different signs, the extended rescaled Poincaré map is associated with a central model. We estimate such central model and show it must have chain recurrent central segments over the singularities. This obstructs the application of central model to create horseshoes, and indicates that, differing from $C^1$ diffeomorphisms, solo using central model method is insufficient as a strategy to prove weak Palis conjecture for higher dimensional ($\geq 4$) singular…
In 1988, Mayer proved the remarkable fact that infinity is an explosion point for the set of endpoints of the Julia set of an exponential map that has an attracting fixed point. That is, the set is totally separated (in particular, it does not have any nontrivial connected subsets), but its union with the point at infinity is connected. Answering a question of Schleicher, we extend this result to the set of "escaping endpoints" in the sense of Schleicher and Zimmer, for any exponential map for which the singular value belongs to an attracting or parabolic basin, has a finite orbit, or escapes …
