A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$
Carlos Gutierrez Nguyen Van Chau
Discrete & Continuous Dynamical Systems - A 2007, 17(2): 397-402 doi: 10.3934/dcds.2007.17.397
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
Simple umbilic points on surfaces immersed in $\R^4$
Carlos Gutierrez Víctor Guíñez
Discrete & Continuous Dynamical Systems - A 2003, 9(4): 877-900 doi: 10.3934/dcds.2003.9.877
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
Discrete & Continuous Dynamical Systems - A 2006, 16(2): 455-461 doi: 10.3934/dcds.2006.16.455
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
Discrete & Continuous Dynamical Systems - A 2010, 26(1): 251-263 doi: 10.3934/dcds.2010.26.251
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
Quartic differential forms and transversal nets with singularities
Carlos Gutierrez Víctor Guíñez Alvaro Castañeda
Discrete & Continuous Dynamical Systems - A 2010, 26(1): 225-249 doi: 10.3934/dcds.2010.26.225
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
Discrete & Continuous Dynamical Systems - A 2007, 17(2): 247-258 doi: 10.3934/dcds.2007.17.247
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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