Generic chaos
Generic diffeomorphisms on compact surfaces
We discuss the remaining obstacles to prove Smale's conjecture about the C¹-density of hyperbolicity among surface diffeomorphisms. Using a C¹-generic approach, we classify the possible pathologies that may obstruct the C¹-density of hyperbolicity. We show that there are essentially two types of obstruction: (i) persistence of infinitely many hyperbolic homoclinic classes and (ii) existence of a single homoclinic class which robustly exhibits homoclinic tangencies. In the course of our discussion,...
Generic linear cocycles over a minimal base
We prove that a generic linear cocycle over a minimal base dynamics of finite dimension has the property that the Oseledets splitting with respect to any invariant probability coincides almost everywhere with the finest dominated splitting. Therefore the restriction of the generic cocycle to a subbundle of the finest dominated splitting is uniformly subexponentially quasiconformal. This extends a previous result for SL(2,ℝ)-cocycles due to Avila and the author.
Generic measures for geodesic flows on nonpositively curved manifolds
We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by trajectories not bounding a flat strip. This is done by showing that Dirac measures on periodic orbits are dense in that set.In the case of a compact surface, we get the following sharp result:...
Generic Nekhoroshev theory without small divisors
In this article, we present a new approach of Nekhoroshev’s theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak, it combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new results of...
Generic observability of dynamical systems.
Generic one-parameter families of vector fields on two-dimensional manifolds
Generic points in the cartesian powers of the Morse dynamical system
The symbolic dynamical system associated with the Morse sequence is strictly ergodic. We describe some topological and metrical properties of the Cartesian powers of this system, and some of its other self-joinings. Among other things, we show that non generic points appear in the fourth power of the system, but not in lower powers. We exhibit various examples and counterexamples related to the property of weak disjointness of measure preserving dynamical systems.
Generic properties of closed geodesics on smooth hypersurfaces.
Generic Properties of Continuous Iterated Function Systems with Place Dependent Probabilities
It is shown that for a typical continuous learning system defined on a compact convex subset of ℝⁿ the Hausdorff dimension of its invariant measure is equal to zero.
Generic Properties of Geodesic Flows.
Generic properties of learning systems
It is shown that the set of learning systems having a singular stationary distribution is generic in the family of all systems satisfying the average contractivity condition.
Generic properties of the rotation number of one-parameter diffeomorphisms of the circle
Generic robustness of spectral decompositions
Generic saddle-node bifurcation for cascade second order ODEs on manifolds
Cascade second order ODEs on manifolds are defined. These objects are locally represented by coupled second order ODEs such that any solution of one of them can represent an external force for the other one. A generic saddle-node bifurcation theorem for 1-parameter families of cascade second order ODEs is proved.
Generic uniqueness of a minimal solution for variational problems on a torus.
Generic uniqueness of minimal configurations with rational rotation numbers in Aubry-Mather theory.
Généricité d'exposants de Lyapunov non-nuls pour des produits déterministes de matrices
Genericity of nonsingular transformations with infinite ergodic index
It is shown that in the group of invertible measurable nonsingular transformations on a Lebesgue probability space, endowed with the coarse topology, the transformations with infinite ergodic index are generic; they actually form a dense set. (A transformation has infinite ergodic index if all its finite Cartesian powers are ergodic.) This answers a question asked by C. Silva. A similar result was proved by U. Sachdeva in 1971, for the group of transformations preserving an infinite measure. Exploring...
Geodätischer Fluß auf Mannigfaltigkeiten vom hyperbolischen Typ.