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Register a new userLinearization of complex hyperbolic Dulac germs
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We prove that a hyperbolic Dulac germ with complex coefficients in its expansion is linearizable on a standard quadratic domain and that the linearizing coordinate is again a complex Dulac germ. The proof uses results about normal forms of hyperbolic transseries from another work of the authors.
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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.
Given a discrete subgroup $Gamma$ of $PU(1,n)$ it acts by isometries on the unit complex ball $Bbb{H}^n_{Bbb{C}}$, in this setting a lot of work has been done in order to understand the action of the group. However when we look at the action of $Gamma$ on all of $ Bbb{P}^n_{Bbb{C}}$ little or nothing is known, in this paper study the action in the whole projective space and we are able to show that its equicontinuity agree with its Kulkarni discontuity set. Morever, in the non-elementary case, this set turns out to be the largest open set on which the group acts properly and discontinuously and can be described as the complement of the union of all complex projective hyperplanes in $ Bbb{P}^n_{Bbb{C}}$ which are tangent to $partial Bbb{H}^n_{Bbb{C}}$ at points in the Chen-Greenberg limit set $Lambda_{CG}(Gamma)$.
Let $R$ be a ring of characteristic $0$ with field of fractions $K$, and let $mge2$. The Bottcher coordinate of a power series $varphi(x)in x^m + x^{m+1}R[![x]!]$ is the unique power series $f_varphi(x)in x+x^2K[![x]!]$ satisfying $varphicirc f_varphi(x) = f_varphi(x^m)$. In this paper we study the integrality properties of the coefficients of $f_varphi(x)$, partly for their intrinsic interest and partly for potential applications to $p$-adic dynamics. Results include: (1) If $p$ is prime and $R=mathbb Z_p$ and $varphi(x)in x^p + px^{p+1}R[![x]!]$, then $f_varphi(x)in R[![x]!]$. (2) If $varphi(x)in x^m + mx^{m+1}R[![x]!]$, then $f_varphi(x)=xsum_{k=0}^infty a_kx^k/k!$ with all $a_kin R$. (3) In (2), if $m=p^2$, then $a_kequiv-1pmod{p}$ for all $k$ that are powers of $p$.
We consider holomorphic semicocycles on the open unit ball in a Banach space taking values in a Banach algebra. We establish criteria for a semicocycle to be linearizable, that is, cohomologically equivalent to one independent of the spatial variable.
We consider holomorphic maps defined in an annulus around $mathbb R/mathbb Z$ in $mathbb C/mathbb Z$. E. Risler proved that in a generic analytic family of such maps $f_zeta$ that contains a Brjuno rotation $f_0(z)=z+alpha$, all maps that are conjugate to this rotation form a codimension-1 analytic submanifold near $f_0$. In this paper, we obtain the Rislers result as a corollary of the following construction. We introduce a renormalization operator on the space of univalent maps in a neighborhood of $mathbb R/mathbb Z$. We prove that this operator is hyperbolic, with one unstable direction corresponding to translations. We further use a holomorphic motions argument and Yoccozs theorem to show that its stable foliation consists of diffeomorphisms that are conjugate to rotations.
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