HOMOCLINIC CLASSES AND FINITUDE OF ATTRACTORS FOR VECTOR FIELDS ON $n$ -MANIFOLDS

Carballo, C. M.; Morales, C. A.

doi:10.1112/s0024609302001601

Cited by 9 publications

(7 citation statements)

References 9 publications

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“…In such examples the classes are either (a) contained in the closure of an infinite set of sinks or sources, or else (b) obtained by considering the product of a three dimensional diffeomorphism f exhibiting a wild class accumulated by (say) sinks by a strong expansion. In case (b) one obtains a diffeomorphism F having a normally hyperbolic wild homoclinic class accumulated by saddles (in this case the dimension of the ambient manifold is at least 4), see [BD 2 ,BDP,CM]. But the existence of wild homoclinic classes accumulated only by infinitely many true saddles (i.e., not obtained via a product) remains an open problem.…”

Section: Is σ the Hausdorff Limit Of Periodic Orbits Of Index τ ? Or mentioning

confidence: 99%

Periodic points and homoclinic classes

Abdenur¹,

Bonatti²,

Crovisier³

et al. 2006

Ergod. Th. Dynam. Sys.

103

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We prove that there is a residual subset I of Diff 1 (M) such that any homoclinic class of a diffeomorphism f ∈ I having saddles of indices α and β contains a dense subset of saddles of index τ for every τ ∈ [α, β] ∩ N. We also derive some consequences from this result about the Lyapunov exponents of periodic points and the sort of bifurcations inside homoclinic classes of generic diffeomorphisms.

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Section: Is σ the Hausdorff Limit Of Periodic Orbits Of Index τ ? Or mentioning

confidence: 99%

Periodic points and homoclinic classes

Abdenur¹,

Bonatti²,

Crovisier³

et al. 2006

Ergod. Th. Dynam. Sys.

103

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“…We remark that examples in [3] and [8] respectively show that neither alternative (b) nor alternative (c) can be removed from the statement of Corollary B.1 for dimensions n 4. Alternative (a) of course cannot be removed due to the existence of omega-stable diffeomorphisms.…”

Section: Statement Of the Resultsmentioning

confidence: 90%

“…Homoclinic classes are the natural candidates to replace hyperbolic basic sets in nonhyperbolic theory. Several recent papers (including Diaz, Pujals and Ures [10], Bonatti, Diaz and Pujals [5], Carballo and Morales [8], and Carballo, Morales and Pacifico [9]) explore their "hyperbolic-like" properties, many of which hold only for generic diffeomorphisms. This paper adopts a similar viewpoint.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…(The actual statement in [5], regarding generic diffeomorphisms that are "far from Newhouse's phenomenon", looks less general, but is easily seen to be equivalent to the statement in the previous sentence.) The example in [8], of a locally residual subset of Diff 1 (M ) where there are only 3 sinks/sources but infinitely many homoclinic classes (and no spectral decomposition), gives a negative answer to this question. But we can obtain a C 1 -generic trichotomy in high dimensions by adding a third alternative:…”

Section: Statement Of the Resultsmentioning

confidence: 99%

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Generic robustness of spectral decompositions

Abdenur

2003

Annales Scientifiques de l’École Normale Supérieure

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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 nonwandering set Ω(f) admits a partition into a finite number of transitive compact sets), then this spectral decomposition is robust in a generic sense (tame behavior). This implies a C 1-generic trichotomy that generalizes some aspects of a two-dimensional theorem of Mañé [Topology 17 (1978) 386-396]. Lastly, Palis [Astérisque 261 (2000) 335-347] has conjectured that densely in Diff k (M) diffeomorphisms either are hyperbolic or exhibit homoclinic bifurcations. We use the aforementioned results to prove this conjecture in a large open region of Diff 1 (M).  2003 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Nous montrons qu'étant donnée une variété compacte M de dimension n, n 2, il existe un sous-ensemble résiduel R de Diff 1 (M) tel que si f ∈ R admet une décomposition spectrale (c'est-à-dire, Ω(f) admet une partition en un nombre fini d'ensembles compacts transitifs), alors cette décomposition spectrale est robuste dans un sens générique. Cela implique une trichotomie générique qui généralise certains aspects d'un théorème bi-dimensionnel de Mañé. Enfin, Palis a conjecturé que dans un sous-ensemble dense de Diff 1 (M), les difféomorphismes ou bien sont hyperboliques, ou bien admettent des bifurcations homoclines. Nous utilisons les résultats précédents pour prouver cette conjecture dans une grande région ouverte de Diff 1 (M).

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“…Theorem 25 [Carballo and Morales 2003]. If f is a C 1 -generic tame diffeomorphism then the union of the basins of its attractors is an open and dense subset of M.…”

Section: Another Source Of Lyapunov Stable Sets Is the Following Whimentioning

confidence: 99%

Untitled

2015

Involve

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The calculation of the probability of correct selection (PCS) shows how likely it is that the populations chosen as "best" truly are the top populations, according to a well-defined standard. PCS is useful for the researcher with limited resources or the statistician attempting to test the quality of two different statistics. This paper explores the theory behind two selection goals for PCS, G-best and d-best, and how they improve previous definitions of PCS for massive datasets. This paper also calculates PCS for two applications that have already been analyzed by multiple testing procedures in the literature. The two applications are in neuroimaging and econometrics. It is shown through these applications that PCS not only supports the multiple testing conclusions but also provides further information about the statistics used.ENHANCING MULTIPLE TESTING 185 selection of the top populations. After an experiment, one can use these methods to calculate the probability that the researcher has found the actual top t populations.If our previous toy example was an actual experiment, we might have a need to find the top, say, one statistic, but we have the resources to study two. We would then use G-best selection, where we choose a fixed amount of populations, t + G, that contains the top t statistics. On the other hand, if we simply needed the populations to be within a certain threshold of quality, we would use d-best selection. In d-best selection, we are finding a random number of populations, say r , which contains populations that are within a certain distance d from the top t populations. The number r is determined by an interval of prespecified length d.Definition 2.1. Let s be the set of the indices corresponding to the top t +G statistics for some prespecified G. Let A t be a set of indices of the top t parameters. ThenA set s that satisfies CS G,t is called G-best, and the probability that we have chosen a G-best set is denoted by P(CS G,t ). 190ERIN IRWIN AND JASON WILSON t 1 2 3 4 P(CS 0,t ) = P( 0 CS t ) 0.11 0.02 0.00 0.00 P(CS 2,t ) 0.33 0.11 0.03 0.01 P(CS 4,t ) 0.49 0.22 0.08 0.03 P(CS 6,t ) 0.60 0.33 0.15 0.06 P( 0.5 CS t ) 0.19 0.04 0.05 0.03 P( 1 CS t ) 0.24 0.19 0.35 0.29P(CS 0,10 ) = 0.13 P( 0 CS 10 ) = 0.13 P(CS 0,10 ) = 0.71 P( 0 CS 10 ) = 0.71 P(CS 1,9 ) = 0.29 P( 0.5 CS 10 ) = 0.32 P(CS 1,9 ) = 0.98 P( 0.5 CS 10 ) = 0.71 P(CS 2,8 ) = 0.53 P( 1 CS 10 ) = 0.52 P(CS 2,8 ) = 1.00 P( 1 CS 10 ) = 0.95 P(CS 3,7 ) = 0.74 P( 1.5 CS 10 ) = 0.78 P(CS 3,7 ) = 1.00 P( 1.5 CS 10 ) = 0.95 P(CS 4,6 ) = 0.92 P( 2 CS 10 ) = 0.86 P(CS 4,6 ) = 1.00 P( 2 CS 10 ) = 0.97 P(CS 5,5 ) = 1.00 P( 2.5 CS 10 ) = 0.93 P(CS 5,5 ) = 1.

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HOMOCLINIC CLASSES AND FINITUDE OF ATTRACTORS FOR VECTOR FIELDS ON $n$ -MANIFOLDS

Cited by 9 publications

References 9 publications

Periodic points and homoclinic classes

Periodic points and homoclinic classes

Generic robustness of spectral decompositions

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