User:Genericity

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: systems with the assumption that the tangent bundle over the limit set $$L(f)$$ splits into two complementary subbundles $$T_{L(f)}M=E^s\oplus E^u $$

It appeared however that hyperbolic systems are in general not dense in the space of dynamics. People therefore looked for a list of few simple phenomena that are dense in the complement of hyperbolic dynamics.

Here is a weaker notion of hyperbolicity: an invariant set $$K$$ has a dominated splitting if its tangent bundle decomposes as a sum $$T_K=E\oplus F$$ of two linear subbundles and if there exists $$N\geq 1$$ such that...