User:Genericity

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For a long time (mainly after Poincar\'e) it has been a goal in the theory of dynamical systems to describe the dynamics from the generic viewpoint, that is, describing the dynamics of ``big sets'' (residual, dense, etc.) of the space of all dynamical systems.

It was briefly thought in the sixties that this could be realized by the so-called hyperbolic ones (See Principal structures (Hasselblatt-Katok) Handbook volume 1A): systems with the assumption that the tangent bundle over the limit set $L(f)$ (see definition in subsection \ref{spec}) splits into two complementary subbundles $T_{L(f)}M=E^s\oplus E^u$ so that vectors in $E^s$ (respectively $E^u$) are uniformly forward (respectively backward) contracted by the tangent map $Df.$ Under this assumption, it is proved that the limit set decomposes into a finite number of transitive sets such that the asymptotic behavior of any orbit is described by the dynamics in the trajectories in those finitely many transitive sets. Moreover, this topological description allows to get the statistical behavior of the system. In other words, hyperbolic dynamics on the tangent bundle characterizes the dynamics over the manifold from a geometrical topological and statistical point of view.

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