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Generic robustness of spectral decompositions. (English. French summary) Zbl 1027.37010
Let $$f\in \text{Diff}^1(M)$$, where $$M$$ is a compact boundaryless manifold of dimension $$n\geq 2$$ and $$p\in M$$ is a periodic hyperbolic saddle of $$f$$. The homoclinic class of $$f$$ relative to $$p$$ is given by $$H(p,f)= \text{cl}[W^s(p)\pitchfork W^u(p)]$$, where $$\pitchfork$$ denotes points of transverse intersection of the invariant manifolds. Homoclinic classes are the natural candidates to replace hyperbolic basic sets in nonhyperbolic theory.
The main result of the paper says that there exists a residual subset $$R$$ of $$\text{Diff}^1(M)$$ such that if $$f\in R$$ has only finitely many distinct homoclinic classes $$H(p_1,f),\dots, H(p_k,f)$$, then there exists a neighborhood $$U$$ of $$f$$ in $$R$$ such that if $$g\in U$$, then the only distinct homoclinic classes of $$g$$ are the continuations $$H(p^g_1, g),\dots, H(p^g_k, g)$$ of the homoclinic classes of $$f$$. Further there exists a residual subset $$\widetilde R$$ of $$\text{Diff}^1(M)$$ such that if $$f\in\widetilde R$$ admits a spectral decomposition, then this one is generically robust. The aforementioned results are used to prove the Palis conjecture, which states that there exists a dense subset $$D$$ such that if $$f\in D$$, then $$f$$ either is hyperbolic or exhibits homoclinic bifurcations.

##### MSC:
 37C05 Dynamical systems involving smooth mappings and diffeomorphisms 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics 37G30 Infinite nonwandering sets arising in bifurcations of dynamical systems 37C29 Homoclinic and heteroclinic orbits for dynamical systems
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