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Let $$\mathrm{Diff }^1(M)$$ Diff 1 ( M ) be the set of all $$C^1$$ C 1 -diffeomorphisms $$f:M\rightarrow M$$ f : M → M , where $$M$$ M is a compact boundaryless d-dimensional manifold, $$d\ge 2$$ d ≥ 2 . We prove that there is a residual subset $$\mathfrak R $$ R of $$\mathrm{Diff }^1(M)$$ Diff 1 ( M ) such that if $$f\in \mathfrak R $$ f ∈ R and if $$H(p)$$ H ( p ) is the homoclinic class associated with a hyperbolic periodic point $$p$$ p , then either $$H(p)$$ H ( p ) admits a dominated splitting of the form $$E\oplus F_1\oplus \dots \oplus F_k\oplus G$$ E ⊕ F 1 ⊕ … ⊕ F k ⊕ G , where $$F_i$$ F i is not hyperbolic and one-dimensional, or $$f|_{H(p)}$$ f | H ( p ) has no symbolic extensions.
Mathematische Zeitschrift – Springer Journals
Published: May 17, 2013
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