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### Open Access Journals

DCDS

In this article we consider the general problem of translating
definitions and results from
the category of discrete-time dynamical systems to the category of flows.
We consider the dynamics of homeomorphisms and flows on compact metric spaces,
in particular Peano continua.
As a translating tool, we construct continuous, symmetric and monotonous
fields of local cross sections for an arbitrary flow without singular points.
Next, we use this structure in the study of expansive flows on Peano continua.
We show that expansive flows have not stable points and
that every point contains a non-trivial continuum in its stable set.
As a corollary we obtain that no
Peano continuum with an open set homeomorphic to the plane
admits an expansive flow.
In particular, compact surfaces do not admit expansive flows without singular points.

DCDS

We prove that the genus two surface admits a cw-expansive homeomorphism
with a fixed point whose local stable set is not locally connected.

DCDS

We give sufficient conditions for a diffeomorphism of a compact surface
to be robustly $N$-expansive and cw-expansive in the $C^r$-topology.
We give examples on the genus two surface showing that they need not to be Anosov diffeomorphisms.
The examples are axiom A diffeomorphisms with tangencies at wandering points.

DCDS

We extend some known results from smooth dynamical systems to
the category of Lipschitz homeomorphisms of compact metric spaces.
We consider dynamical properties as robust expansiveness and structural stability allowing Lipschitz perturbations
with respect to a hyperbolic metric.
We also study the relationship between Lipschitz topologies and the $C^1$ topology on smooth manifolds.

DCDS

We prove that a flow without singular points of index zero on a compact surface is expansive if and only if the singularities
are of saddle type and the union of their separatrices is dense.
Moreover we show that such flows are obtained by surgery on the suspension of minimal interval exchange maps.

DCDS

We study continuum-wise expansive flows with fixed points on metric spaces and low dimensional manifolds. We give sufficient conditions for a surface flow to be singular cw-expansive and examples showing that cw-expansivity does not imply expansivity. We also construct a singular Axiom A vector field on a three-manifold being singular cw-expansive and with a Lorenz attractor and a Lorenz repeller in its non-wandering set.

keywords:
Singular Axiom A
,
expansive flows
,
flows on surfaces
,
continuum theory
,
Lorenz attractor

## Year of publication

## Related Authors

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