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Register a new userA two-dimensional polynomial mapping with a wandering Fatou component
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We show that there exist polynomial endomorphisms of C^2, possessing a wandering Fatou component. These mappings are polynomial skew-products, and can be chosen to extend holomorphically of P^2(C). We also find real examples with wandering domains in R^2. The proof is based on parabolic implosion techniques, and is based on an original idea of M. Lyubich.
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Let T be a finite subset of the complex unit circle S^1, and define f: S^1 -> S^1 by f(z) = z^d. Let CH(T) denote the convex hull of T. If card(T) = N > 2, then CH(T) defines a polygon with N sides. The N-gon CH(T) is called a emph{wandering N-gon} if for every two non-negative integers i eq j, CH(f^i(T)) and CH(f^j(T)) are disjoint N-gons. A non-degenerate chord of S^1 is said to be emph{critical} if its two endpoints have the same image under f. Then for a critical chord, it is natural to define its (forward) orbit by the forward iterates of the endpoints. Similarly, call a critical chord emph{recurrent} if one of its endpoints is recurrent under f. The main result of our study is that a wandering N-gon has at least N-1 recurrent critical chords in its limit set (defined in a natural way) having pairwise disjoint, infinite orbits. Using this result, we are able to strengthen results of Blokh, Kiwi and Levin about wandering polygons of laminations. We also discuss some applications to the dynamics of polynomials. In particular, our study implies that if v is a wandering non-precritical vertex of a locally connected polynomial Julia set, then there exists at least ord(v)-1 recurrent critical points with pairwise disjoint orbits, all having the same omega-limit set as v. Thus, we likewise strengthen results about wandering vertices of polynomial Julia sets.
By applying holomorphic motions, we prove that a parabolic germ is quasiconformally rigid, that is, any two topologically conjugate parabolic germs are quasiconformally conjugate and the conjugacy can be chosen to be more and more near conformal as long as we consider these germs defined on smaller and smaller neighborhoods. Before proving this theorem, we use the idea of holomorphic motions to give a conceptual proof of the Fatou linearization theorem. As a by-product, we also prove that any finite number of analytic germs at different points in the Riemann sphere can be extended to a quasiconformal homeomorphism which can be more and more near conformal as as long as we consider these germs defined on smaller and smaller neighborhoods of these points.
We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an unlikely intersection statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dynamical Andre-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.
We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where $k$ critical points bifurcate emph{independently}, with $k$ up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an analytical criterion for the non-vanishing of the self-intersections of the bifurcation current, we deduce the equality of the supports of the bifurcation current and the bifurcation measure for such families. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior.As part of our proof we construct, in these families, subfamilies of codimension 1 where the bifurcation locus has non empty interior. This provides a new independent proof of the existence of holomorphic families of arbitrarily large dimension whose bifurcation locus has non empty interior. Finally, it shows that the Hausdorff dimension of the support of the bifurcation measure is maximal at any point of its support.
For every $minmathbb{N}$, we establish the equidistribution of the sequence of the averaged pull-backs of a Dirac measure at any given value in $mathbb{C}setminus{0}$ under the $m$-th order derivatives of the iterates of a polynomials $fin mathbb{C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of $f$ with pole at $infty$. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field $k$ for a sequence of effective divisors on $mathbb{P}^1(overline{k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Henon-type polynomial automorphism of $mathbb{C}^2$ has a given eigenvalue.
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