## Journals

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### Open Access Journals

DCDS

We study local problems around simple umbilic points of surfaces immersed in $\mathbb R^4$
such as finite determinacy and versal unfoldings.

DCDS

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.

DCDS

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.

DCDS

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.
DCDS

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.

DCDS

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.

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