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Guillermo Sienra

Publications and source records attributed to Guillermo Sienra.

7 recordsLinked to original sources
arXiv2025

Structural stability in piecewise Möbius transformations

Structural stability of piecewise Möbius transformations (PMTs) is examined from various perspectives. A result concerning structural stability, restricted to the space of PMTs, is derived using hyperbolic characteristics of the component functions and the pre-singularities set, which facilitates a holomorphic motion. The analogous concept of J-stability for rational maps is defined and analyzed for PMTs, revealing some connections to general structural stability. The definitions of hyperbolic and expansive PMTs are introduced, demonstrating that they are not equivalent and that neither implies structural stability. By synthesizing the previous results and analyses, sufficient conditions for structural stability are established. Lastly, an example of structural stability within the tent maps family, extended to the complex plane, is presented.

math.DS↗
arXiv2019

On hyperbolic cobordisms and Hurwitz classes of holomorphic coverings

In this article we show that for every collection $\mathcal{C}$ of an even number of polynomials, all of the same degree $d>2$ and in general position, there exist two hyperbolic $3$-orbifolds $M_1$ and $M_2$ with a Möbius morphism $α:M_1\rightarrow M_2$ such that the restriction of $α$ to the boundaries $\partial M_1$ and $\partial M_2$ forms a collection of maps $Q$ in the same conformal Hurwitz class of the initial collection $\mathcal{C}$. Also, we discuss the relationship between conformal Hurwitz classes of rational maps and classes of continuous isomorphisms of sandwich products on the set of rational maps.

math.DS↗
arXiv2019

Dynamical aspects of piecewise conformal maps

We study the dynamics of piecewise conformal maps in the Riemann sphere. The normality and chaotic regions are defined and we state several results and properties of these sets. We show that the stability of these piecewise maps is related to the Kleinian group generated by their transformations under certain hypotheses. The general motivation of the article is to compare the dynamics of piecewise conformal maps and those of the Kleinian groups and iterations of rational maps.

math.DS↗
arXiv2013

On Poincaré extensions of rational maps

There is a classical extension, of Möbius automorphisms of the Riemann sphere into isometries of the hyperbolic space $\mathbb{H}^3$, which is called the Poincaré extension. In this paper, we construct extensions of rational maps on the Riemann sphere over endomorphisms of $\mathbb{H}^3$ exploiting the fact that any holomorphic covering between Riemann surfaces is Möbius for a suitable choice of coordinates. We show that these extensions define conformally natural homomorphisms on suitable subsemigroups of the semigroup of Blaschke maps. We extend the complex multiplication to a product in $\mathbb{H}^3$ that allows to construct a visual extension of any given rational map.

math.DS↗
arXiv2010

Univalent Baker domains and boundary of deformations

For $f$ an entire transcendental map with a univalent Baker domain $U$ of hyperbolic type I, we study pinching deformations in $U$, the support of this deformation being certain laminations in the grand orbit of $U$. We show that pinching along a lamination that contains the geodesic $λ_{\infty}$ (See Section 3.1) does not converges. However, pinching at a lamination that does not contains such $λ_{\infty}$, converges and converges to a unique map $F$ if: the Julia set of $f$, $J(f)$ is connected, the postcritical set of $f$ is a positive (plane) distance away from $J(f)$, and it is thin at $\infty$. We show that $F$ has a simply connected wandering domain that stays away from the postcritical set. We interpret these results in terms of the Teichmüller space of $f$, $Teich(f)$, included in $M_{f}$ the marked space of topologically equivalent maps to $f$.

math.DS↗
arXiv2005

Poincare Series and instability of exponential maps

We relate the properties of the postsingular set for the exponential family to the questions of stability. We calculate the action of the Ruelle operator for the exponential family. We prove that if the asymptotic value is a summable point and its orbit satisfies certain topological conditions, the map is unstable hence there are no Beltrami differentials in the Julia set. Also we show that if the postsingular set is a compact set, then the singular value is summable.

math.DS↗