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Liz Vivas

Publications and source records attributed to Liz Vivas.

15 recordsLinked to original sources
arXiv2026

An arithmetic approach to parabolic multiplicity in complex dynamics

When $\omega$ is a primitive $n$-th root of unity, the quadratic polynomial $F(z) = \omega z (1 -z)$ and the entire map $F(z) = \omega z \mathrm{e}^{-z}$ both have a parabolic fixed point at $0$. Their parabolic multiplicity is equal to $1$, that is, $F^{\circ n}(z) = z \bigl( 1 +c z^n +\mathcal{O}(z^{n+1}) \bigr)$ with $c \neq 0$. The classical proof of this fact is transcendental. We present an arithmetic proof which may be extracted from [Towards global models near homoclinic tangencies of dissipative diffeomorphisms; H. Broer, C. Sim\'{o}, J.C. Tatjer] in the transcendental case and requires working in $\mathbb{Z}/(n -1) \mathbb{Z}$, and which is new in the polynomial case and requires working in the $p$-adic field $\mathbb{Q}_p$ for a suitable prime $p$ such that the order of $2$ in $(\mathbb{Z}/p \mathbb{Z})^\times$ is exactly $n$.

math.DS↗
arXiv2026

A Non-Autonomous Model for Parabolic Implosion

Orthogonal polynomials appear naturally in the study of compositions of M\"obius transformations. In this paper, we consider several classes of orthogonal polynomials associated to non-autonomous perturbations of a parabolic M\"obius map. Our results can be viewed as instances of non-autonomous parabolic implosion, including a random perturbative regime in which convergence holds almost surely.

math.CV↗
arXiv2022

On the dimension of bundle-valued Bergman spaces on compact Riemann surfaces

Given a holomorphic vector bundle $E$ over a compact Riemann surface $M$, and an open set $D$ in $M$, we prove that the Bergman space of holomorphic sections of the restriction of $E$ to $D$ must either coincide with the space of global holomorphic sections of $E$, or be infinite dimensional. Moreover, we characterize the latter entirely in terms of potential-theoretic properties of $D$.

math.CV↗
arXiv2022

Stable manifolds of biholomorphisms in $\mathbb{C}^n$ asymptotic to formal curves

Given a germ of biholomorphism $F\in\mathrm{Diff}(\mathbb{C}^n,0)$ with a formal invariant curve $Γ$ such that the multiplier of the restricted formal diffeomorphism $F|_Γ$ is a root of unity or satisfies $|(F|_Γ)'(0)|<1$, we prove that either $Γ$ is contained in the set of periodic points of $F$ or there exists a finite family of stable manifolds of $F$ where all the orbits are asymptotic to $Γ$ and whose union eventually contains every orbit asymptotic to $Γ$. This result generalizes to the case where $Γ$ is a formal periodic curve.

math.DS↗
arXiv2021

On the dimension of Bergman spaces on $\mathbb{P}^1$

Inspired by a result by Sz\H{o}ke, we give potential-theoretic characterizations of the dimension of the Bergman space of holomorphic sections of a restriction of a holomorphic line bundle of $\mathbb{P}^1$ to some open set $D\subset\mathbb{P}^1$.

math.CV↗
arXiv2020

Hardy Spaces for a Class of Singular Domains

We set a framework for the study of Hardy spaces inherited by complements of analytic hypersurfaces in domains with a prior Hardy space structure. The inherited structure is a filtration, various aspects of which are studied in specific settings. For punctured planar domains, we prove a generalization of a famous rigidity lemma of Kerzman and Stein. A stabilization phenomenon is observed for egg domains. Finally, using proper holomorphic maps, we derive a filtration of Hardy spaces for certain power-generalized Hartogs triangles, although these domains fall outside the scope of the original framework.

math.CV↗
arXiv2019

Non-autonomous Parabolic Bifurcation

Let $f(z) = z+z^2+O(z^3)$ and $f_ε(z) = f(z) + ε^2$. A classical result in parabolic bifurcation in one complex variable is the following: if $N-\fracπε\to 0$ we obtain $(f_ε)^{N} \to \mathcal{L}_f$, where $\mathcal{L}_f$ is the Lavaurs map of $f$. In this paper we study a \textit{non-autonomous} parabolic bifurcation. We focus on the case of $f_0(z)=\frac{z}{1-z}$. Given a sequence $\{ε_i\}_{1\leq i\leq N}$, we denote $f_n(z) = f_0(z) + ε_n^2$. We give sufficient and necessary conditions on the sequence $\{ε_i\}$ that imply that $f_{N}\circ\ldots f_{1} \to \textrm{Id}$ (the Lavaurs map of $f_0$). We apply our results to prove parabolic bifurcation phenomenon in two dimensions for some class of maps.

math.CV↗
arXiv2018

Local dynamics of parabolic skew-products

The local dynamics around a fixed point has been extensively studied for germs of one and several complex variables. In one dimension, there exist a complete picture of the trajectory of the orbits on a whole neighborhood of the fixed point. In dimensions larger or equal than two some partial results are known. In this article we analyze a case that lies in the boundary between one and several complex variables. We consider skew product maps of the form F (z, w) = (λ(z), f (z, w)). We deal with the case of parabolic skew product maps, that is when DF(0,0) = Id. Our goal is to describe the behavior of orbits around a whole neighborhood of the origin. We establish formulas for conjugacy maps in different regions of a neighborhood of the origin.

math.CV↗
arXiv2014

Parametrization of unstable manifolds for parabolic skew-products

Given a parabolic map in one dimension $f(z) = z+O(z^2)$, $f \neq Id$, it is known that there exists the analogous of stable and unstable domains. That is, domains in which every point is attracted by $f$ (and by the inverse $f^{-1}$) towards the fixed point. In this paper we prove that there exists a natural parametrization for the unstable manifold in terms of iterates for some subset of parabolic maps. Furthermore, we prove that this parametrization is valid also in the case of skew-product maps that satisfy certain conditions. Finally, we give an application of this fact to construct Fatou disks for skew-product maps that are parabolic in each direction.

math.CV↗
arXiv2014

A survey on non-autonomous basins in several complex variables

Consider a holomorphic automorphism which acts hyperbolically on some invariant compact set. Then for every point in the compact set there exists a stable manifold, which is a complex manifold diffeomorphic to real Euclidean space. If the point is fixed, then the stable manifold is even biholomorphic to complex Euclidean space. In fact, it is known that the stable manifold of a generic point is biholomorphic to Euclidean space, and it has been conjectured that this holds for every point. In this article we survey the history of this problem, addressing both known results and the techniques used to obtain those results. Moreover, we present a list of seemingly simpler open problems and prove several new results, all pointing towards a positive answer to the conjecture discussed above.

math.CV↗
arXiv2013

Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics

Using Green's hyperplane restriction theorem, we prove that the rank of a Hermitian form on the space of holomorphic polynomials is bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a rigidity theorem for CR mappings between hyperquadrics in the spirit of the results of Baouendi-Huang and Baouendi-Ebenfelt-Huang. Given a real-analytic CR mapping of a hyperquadric (not equivalent to a sphere) to another hyperquadric $Q(A,B)$, either the image of the mapping is contained in a complex affine subspace, or $A$ is bounded by a constant depending only on $B$. Finally, we prove a stability result about existence of nontrivial CR mappings of hyperquadrics. That is, as long as both $A$ and $B$ are sufficiently large and comparable, then there exist CR mappings whose image is not contained in a hyperplane. The rigidity result also extends when mapping to hyperquadrics in infinite dimensional Hilbert-space.

math.CV↗
arXiv2012

Dynamics of two-resonant biholomorphisms

In this paper we study the existence of basins of attraction for germs of 2-resonant biholomorphisms of $\C^n$ fixing a point, that is germs such that the eigenvalues of the differential at the fixed point have a 2 dimensional family of resonances.

math.CV↗
arXiv2011

Degenerate characteristic directions for maps tangent to the Identity

Let F be a germ of (C^2,O) tangent to the identity. Assume F has a characteristic direction [v]. In [Hak] Hakim gives conditions to guarantee the existence of an attracting basin to the origin along [v], in the case of [v] a non-degenerate characteristic direction. In this paper we give conditions to guarantee the existence of basins along [v] in the case of [v] a degenerate characteristic direction.

math.CV↗
arXiv2011

Geodesics in the space of Kähler metrics

Let (X,ω) be a compact Kähler manifold. As discovered in the late 1980s by Mabuchi, the set H_0 of Kähler forms cohomologous to ωhas the natural structure of an infinite dimensional Riemannian manifold. We address the question whether any two points in H_0 can be connected by a smooth geodesic, and show that the answer, in general, is "no".

math.CV↗