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Nombre de rotation et dynamique faiblement hyperbolique.

Abstract : This thesis deals with two main branches of dynamical systems: the rotation number theory for degree-one circle endomorphisms and for annulus twist maps, and the theory of non-uniformly hyperbolic dynamical systems. First we define the almost sure rotation number for some circle endomorphisms. It is the rotation number for almost every point of the circle. We describe it for a family of expanding piecewise affine bimodal endomorphisms. We study its regularity and show that the set of parameters which give an irrational almost sure rotation number has full Lebesgue measure. Then we consider annulus twist maps and more precisely bimodal maps from the fattened Arnol'd family. A key role is played by twist-free orbits. It is shown that the set of maps that possess a given rotation number formes, in the parameter space, a tongue bounded by two surfaces. The boundary of rational tongues is associated with homoclinic and saddle-node bifurcations. We finish with some estimates on the size of the rotation set and on the Birkhoff attractor. The appendix is devoted to saddle-node bifurcations of locally maximal hyperbolic sets whose unstable direction is one-dimensional. This bifurcation preserves the geometrical decomposition of the tangent space into stable and unstable spaces. However the expansion along the unstable direction degenerates near a periodic orbit. The bifuraction is one-codimensional.
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Contributor : Sylvain Crovisier <>
Submitted on : Monday, March 4, 2002 - 1:51:30 PM
Last modification on : Wednesday, April 3, 2019 - 1:35:33 AM
Long-term archiving on: : Friday, April 2, 2010 - 5:56:46 PM


  • HAL Id : tel-00001185, version 1



Sylvain Crovisier. Nombre de rotation et dynamique faiblement hyperbolique.. Mathématiques [math]. Université Paris Sud - Paris XI, 2001. Français. ⟨tel-00001185⟩