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**publisher and metadata sources**).We prove that the shadowing property does not hold for
diffeomorphisms in an open and dense subset of the set of
$C^1$-robustly non-hyperbolic transitive diffeomorphisms (i.e.,
diffeomorphisms… Continue Reading

Abstract We prove that given a compact n -dimensional boundaryless manifold M , n ⩾2, there exists a residual subset R of Diff 1 ( M ) such that if f ∈ R admits a spectral decomposition (i.e., the… Continue Reading

We study the ergodic theory of non-conservative C1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with… Continue Reading

We discuss the remaining obstacles to prove Smale’s conjecture about the C-density of hyperbolicity among surface diffeomorphisms. Using a C-generic approach, we classify the possible pathologies… Continue Reading

1 February 2007

We prove that there is a residual subset $\mathcal{I}$ of ${\rm Diff}^1({\it M})$ such that any homoclinic class of a diffeomorphism $f\in \mathcal{I}$ having saddles of indices $\alpha$ and $\beta$… Continue Reading

We study the ergodic theory of non-conservative C^1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with… Continue Reading

We prove that, on connected compact manifolds, both C 1-generic conservative diffeomorphisms and C 1-generic transitive diffeomorphisms are topologically mixing. This is obtained through a… Continue Reading

We study diffeomorphisms of a closed connected manifold whose non-wandering set has a non-empty interior and conjecture that C1-generic diffeomorphisms whose non-wandering set has a non-empty… Continue Reading

We prove that given a compact n-dimensional boundaryless manifold M, n ≥ 2, there exists a residual subset R of the space of C 1 difieomorphisms Diff 1 (M) such that given any chain-transitive set K… Continue Reading

We prove that nontrivial homoclinic classes of C r -generic flows are topologically mixing. This implies that given A, a nontrivial C 1 -robustly transitive set of a vector field X, there is a C 1… Continue Reading