Simple umbilic points on surfaces immersed in $\R^4$
Carlos Gutierrez Víctor Guíñez
We study local problems around simple umbilic points of surfaces immersed in $\mathbb R^4$ such as finite determinacy and versal unfoldings.
keywords: quartic differential forms. umbilic points Lines of curvature smooth immersions
Iterated images and the plane Jacobian conjecture
Ronen Peretz Nguyen Van Chau L. Andrew Campbell Carlos Gutierrez
We show that the iterated images of a Jacobian pair $f:\mathbb{C}^2 \rightarrow \mathbb{C}^2$ stabilize; that is, all the sets $f^k(\mathbb{C}^2)$ are equal for $k$ sufficiently large. More generally, let $X$ be a closed algebraic subset of $\mathbb{C}^N$, and let $f:X\rightarrow X$ be an open polynomial map with $X-f(X)$ a finite set. We show that the sets $f^k(X)$ stabilize, and for any cofinite subset $\Omega \subseteq X$ with $f(\Omega) \subseteq \Omega$, the sets $f^k(\Omega)$ stabilize. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.
keywords: polynomial map etale Jacobian conjecture. Stable image
Transitive circle exchange transformations with flips
Carlos Gutierrez Simon Lloyd Vladislav Medvedev Benito Pires Evgeny Zhuzhoma
We study the existence of transitive exchange transformations with flips defined on the unit circle $S^1$. We provide a complete answer to the question of whether there exists a transitive exchange transformation of $S^1$ defined on $n$ subintervals and having $f$ flips.
keywords: orientation reversing. interval exchange transformation Rauzy induction
A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$
Carlos Gutierrez Nguyen Van Chau
Using the half-Reeb component technique as introduced in [10], we try to clarify the intrinsic relation between the injectivity of differentiable local homeomorphisms $X$ of $R^2$ and the asymptotic behavior of real eigen-values of derivations $DX(x)$. The main result shows that a differentiable local homeomorphism $X$ of $R^2$ is injective and that its image $X(R^2)$ is a convex set if $X$ satisfies the following condition: (*) There does not exist a sequence $R^2$ ∋ $x_i\rightarrow \infty$ such that $X(x_i)\rightarrow a\in \R^2$ and $DX(x_i)$ has a real eigenvalue $\lambda _i\rightarrow 0$. When the graph of $X$ is an algebraic set, this condition becomes a necessary and sufficient condition for $X$ to be a global diffeomorphism.
keywords: eigenvalue condition polynomial diffeomorphism. Injective differentiable maps
Quartic differential forms and transversal nets with singularities
Carlos Gutierrez Víctor Guíñez Alvaro Castañeda
We consider a class $ \mathcal{Q}(M) \,$ consisting of smooth quartic differential forms which are defined on an oriented two-manifold $ M $, to each of which we associate a pair of transversal nets with common singularities. These quartic forms are related to geometric objects such as curvature lines, asymptotic lines of surfaces immersed in $\R^4.$ Local problems around the rank-2 singular points of the elements of $ \mathcal{Q}(M) \,$, such as stability, normal forms, finite determinacy, versal unfoldings, are studied in [2]. Here we make a similar study for a rank-1 singular point that is analogous to the saddle-node singularity of vector fields.
keywords: Quartic Differential Forms nets.
Hopf bifurcation at infinity for planar vector fields
Begoña Alarcón Víctor Guíñez Carlos Gutierrez
We study, from a new point of view, families of planar vector fields without singularities $ \{ X_{\mu}$   :   $-\varepsilon < \mu < \varepsilon\} $ defined on the complement of an open ball centered at the origin such that, at $\mu=0$, infinity changes from repellor to attractor, or vice versa. We also study a sort of local stability of some $C^1$ planar vector fields around infinity.
keywords: Hopf bifurcation vector field Poincaré index. singular points

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