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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-maps having hyperbolic periodic points

N. Aoki, Kazumine Moriyasu, N. Sumi (2001)

Fundamenta Mathematicae

We show that the C¹-interior of the set of maps satisfying the following conditions: (i) periodic points are hyperbolic, (ii) singular points belonging to the nonwandering set are sinks, coincides with the set of Axiom A maps having the no cycle property.

Controllability properties of a class of systems modeling swimming microscopic organisms

Mario Sigalotti, Jean-Claude Vivalda (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a finite-dimensional model for the motion of microscopic organisms whose propulsion exploits the action of a layer of cilia covering its surface. The model couples Newton's laws driving the organism, considered as a rigid body, with Stokes equations governing the surrounding fluid. The action of the cilia is described by a set of controlled velocity fields on the surface of the organism. The first contribution of the paper is the proof that such a system is generically controllable...

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