Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability

Araújo, Vı́tor

doi:10.1017/etds.2020.91

Cited by 11 publications

(13 citation statements)

References 57 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 2. This result is also proven in [5], and is also a consequence of the arguments in both [3] and [16]. We present an additional proof of this result using Theorem 3.1 directly.…”

supporting

confidence: 55%

“…There is also a large body of work on singular hyperbolic and sectionalhyperbolic flows more generally. In [17], it is shown that singular hyperbolic flows admit finitely many ergodic physical measures; more recently, it was shown in [3] that flows of Hölder-C 1 vector fields admitting a sectional-hyperbolic attracting set admit finitely many ergodic SRB measures. The proof in [3] also relies on Poincaré return maps, and so these results extend to discrete singular hyperbolic maps arising as Poincaré maps of hyperbolic flows.…”

mentioning

confidence: 99%

“…In [17], it is shown that singular hyperbolic flows admit finitely many ergodic physical measures; more recently, it was shown in [3] that flows of Hölder-C 1 vector fields admitting a sectional-hyperbolic attracting set admit finitely many ergodic SRB measures. The proof in [3] also relies on Poincaré return maps, and so these results extend to discrete singular hyperbolic maps arising as Poincaré maps of hyperbolic flows. For a detailed discussion of the ergodic properties of hyperbolic flows and their attractors, see [4].…”

mentioning

confidence: 99%

See 2 more Smart Citations

SRB measures of singular hyperbolic attractors

Veconi¹

2022

DCDS

View full text Add to dashboard Cite

<p style='text-indent:20px;'>It is known that hyperbolic maps admitting singularities have at most countably many ergodic Sinai-Ruelle-Bowen (SRB) measures. These maps include the Belykh attractor, the geometric Lorenz attractor, and more general Lorenz-type systems. In this paper, we establish easily verifiable sufficient conditions guaranteeing that the number of ergodic SRB measures is at most finite, and provide examples and nonexamples showing that the conditions are necessary in general.</p>

show abstract

“…Remark 2. This result is also proven in [5], and is also a consequence of the arguments in both [3] and [16]. We present an additional proof of this result using Theorem 3.1 directly.…”

supporting

confidence: 55%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

SRB measures of singular hyperbolic attractors

Veconi¹

2022

DCDS

View full text Add to dashboard Cite

show abstract

“…For item (2), note that for each x ∈ Γ 1 we have that R(x) ∈ ∂Σ j (a 0 ) ⊂ Σ j . Thus there exists a neighborhood…”

Section: 42mentioning

confidence: 99%

On robust expansiveness for sectional hyperbolic attracting sets

Araújo¹,

Junilson²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We prove that sectional-hyperbolic attracting sets for C 1 vector fields are robustly expansive (under an open technical condition of strong dissipativeness for higher codimensional cases). This extends known results of expansiveness for singular-hyperbolic attractors in 3-flows even in this low dimensional setting. We deduce some converse results taking advantage of recent progress in the study of star vector fields: a robustly expansive non-singular vector field is uniformly hyperbolic; and a robustly transitive attractor is sectional-hyperbolic if, and only if, it is robustly expansive. In a low dimensional setting, we show that an attracting set of a 3-flow is singular-hyperbolic if, and only if, it is robustly chaotic (robustly sensitive to initial conditions). CONTENTS 1. Introduction 2. Statement of the results 2.1. Sectional-hyperbolic attracting sets 2.2. Robust expansiveness for codimension-two sectional-hyperbolic attracting sets 2.3. Robust expansiveness for higher codimension 2.4. Some consequences of robust expansiveness 2.5. Organization of the text Acknowledgements 3. Preliminary results on sectional-hyperbolic attracting sets 3.1. Generalized Lorenz-like singularities 3.2. Invariant extension of the stable bundle 3.3. Extension of the center-unstable cone field 3.4. Global Poincaré map on adapted cross-sections 3.5. Hyperbolicity of the global Poincaré map 3.6. Distance between points on distinct stable leaves in cross-sections 3.7. The Poincaré quotient map on a cross-section 4. Proof of robust expansiveness 4.1. Proof of the main Theorems A and B 4.2. Proof of positive expansiveness 4.3. Proof of the claim

show abstract

“…A natural notion of stability for maps with several physical measures supported on a given attracting set was provided in [3]. The same strategy was applied to obtain statistical stability for sectional-hyperbolic attracting sets (a higher (co)dimensional extension of the notion of singular-hyperbolicity) in [8], where the focus lies on the technically much harder task of deducing the properties needed to apply the criteria, due to the high dimensionality of the objects involved.…”

Section: Introductionmentioning

confidence: 99%

On the statistical stability of families of attracting sets and the contracting Lorenz attractor

Araujo

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We present criteria for statistical stability of attracting sets for vector fields using dynamical conditions on the corresponding generated flows. These conditions are easily verified for all singular-hyperbolic attracting sets of C 2 vector fields using known results, providing robust examples of statistically stable singular attracting sets (encompassing in particular the Lorenz and geometrical Lorenz attractors). These conditions are shown to hold also on the persistent but non-robust family of contracting Lorenz flows (also known as Rovella attractors), providing examples of statistical stability among members of non-open families of dynamical systems. In both instances, our conditions avoid the use of detailed information about perturbations of the one-dimensional induced dynamics on specially chosen Poincaré sections. Contents 1. Introduction 1 1.1. Statements of the results 3 1.2. Application to singular-hyperbolic attracting sets 4 1.3. Application to the Contracting Lorenz (Rovella) attractor 6 1.4. Organization of the text 8 2. The contracting Lorenz family of attractors 8 2.1. Existence and uniqueness of physical measure 8 2.2. The physical measure is a SRB measure 9 2.3. Robust expansiveness of contracting Lorenz flows 10 3. Proof of Statistical Stability 11 3.1. Entropy expansiveness 12 3.2. Statistical stability 13 References 13

show abstract

Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability

Cited by 11 publications

References 57 publications

SRB measures of singular hyperbolic attractors

SRB measures of singular hyperbolic attractors

On robust expansiveness for sectional hyperbolic attracting sets

On the statistical stability of families of attracting sets and the contracting Lorenz attractor

Contact Info

Product

Resources

About