[Todos CMAT] Seminario dinámica-Viernes- Pierre Guiraud

[Rafael Potrie]

Hola,

Este viernes 29 a las 14.30 (IMERL) recibimos a Pierre Guiraud (Universidad
de Valparaiso, Chile) que hablará (en español) sobre "On the asymptotic
properties of piecewise contracting maps". Abajo encontraran el resumen.

Son todos bienvenidos,

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Title: On the asymptotic properties of piecewise contracting maps.

Pierre Guiraud, CIMFAV, U. de Valparaíso.

We study   the asymptotic dynamics of maps which are piecewise contracting
on a compact space. These maps are Lipschitz continuous, with Lipschitz
constant smaller than one, when restricted to any piece of a finite and
dense union of disjoint open pieces. We focus on the topological and the
dynamical properties of the (global) attractor of the orbits that remain in
this union. As a starting point, we show that the attractor consists of a
finite set of periodic points when it does not intersect the boundary of a
contraction piece, which complements similar results proved for more
specific classes of piecewise contracting maps. Then, we explore the case
where the attractor intersects these boundaries by providing examples that
show the rich phenomenology of these systems. Due to the discontinuities,
the asymptotic behaviour is not always properly represented by the dynamics
in the attractor. Hence, we introduce generalized orbits to describe the
asymptotic dynamics and its recurrence and transitivity properties. Our
examples include transitive and recurrent attractors, that are either
finite, countable, or a disjoint union of a Cantor set and a countable set.
We also show that the attractor of a piecewise contracting map is usually a
Lebesgue measure-zero set, and we give conditions ensuring that it is
totally disconnected. Finally, we provide an example of piecewise
contracting map with positive topological entropy and whose attractor is an
interval. This is a joint work with E. Catsigeras, A. Meyroneinc and E.
Ugalde.

-- 
Rafael Potrie
rafaelpotrie en gmail.com
http://www.cmat.edu.uy/~rpotrie/
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