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C¹ stability of endomorphisms on two-dimensional manifolds

J. Iglesias, A. Portela, A. Rovella (2012)

Fundamenta Mathematicae

A set of necessary conditions for C¹ stability of noninvertible maps is presented. It is proved that the conditions are sufficient for C¹ stability in compact oriented manifolds of dimension two. An example given by F. Przytycki in 1977 is shown to satisfy these conditions. It is the first example known of a C¹ stable map (noninvertible and nonexpanding) in a manifold of dimension two, while a wide class of examples are already known in every other dimension.

C¹ stable maps: examples without saddles

J. Iglesias, A. Portela, A. Rovella (2010)

Fundamenta Mathematicae

We give here the first examples of C¹ structurally stable maps on manifolds of dimension greater than two that are neither diffeomorphisms nor expanding. It is shown that an Axiom A endomorphism all of whose basic pieces are expanding or attracting is C¹ stable. A necessary condition for the existence of such examples is also given.

C¹-Stably Positively Expansive Maps

Kazuhiro Sakai (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

The notion of C¹-stably positively expansive differentiable maps on closed C ∞ manifolds is introduced, and it is proved that a differentiable map f is C¹-stably positively expansive if and only if f is expanding. Furthermore, for such maps, the ε-time dependent stability is shown. As a result, every expanding map is ε-time dependent stable.

Composite control of the n -link chained mechanical systems

Jiří Zikmund (2008)

Kybernetika

In this paper, a new control concept for a class of underactuated mechanical system is introduced. Namely, the class of n -link chains, composed of rigid links, non actuated at the pivot point is considered. Underactuated mechanical systems are those having less actuators than degrees of freedom and thereby requiring more sophisticated nonlinear control methods. This class of systems includes among others frequently used for the modeling of walking planar structures. This paper presents the stabilization...

Conjugacy of normally tangent diffeomorphisms : a tool for treating moduli of stability

Patrick Bonckaert (1990)

Annales de l'institut Fourier

We give sufficient conditions for the conjugacy of two diffeomorphisms coinciding on a common invariant submanifold V and with equal normal derivative; moreover we obtain that the homeomorphism h realizing this conjugacy satisfies additional inequalities. These inequalities, implying also the existence of the normal derivative of h along V, serve to extend this conjugacy towards regions where moduli of stability are present.

Continuum-wise expansive diffeomorphisms.

Kazuhiro Sakai (1997)

Publicacions Matemàtiques

In this paper, we show that the C1 interior of the set of all continuum-wise expansive diffeomorphisms of a closed manifold coincides with the C1 interior of the set of all expansive diffeomorphisms. And the C1 interior of the set of all continuum-wise fully expansive diffeomorphisms on a surface is investigated. Furthermore, we have necessary and sufficient conditions for a diffeomorphism belonging to these open sets to be Anosov.

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