Robust transitivity of singular hyperbolic attractors

“…The proofs of Theorems A and B use a construction of adapted cross-sections, generalizing that presented in the 3-flow setting in [8] and in the codimension 2 setting in [5], which has been used to prove many delicate statistical properties of these flows; a similar construction (but built in a different way) of higher-dimensional adapted cross-sections was recently proposed in [19]. This enables us to solve the basin problem as follows; see, for example, [13] for a similar but more delicate instance in a highly non-uniformly hyperbolic setting.…”

Section: Araujomentioning

confidence: 99%

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

Araújo¹

2020

Ergod. Th. Dynam. Sys.

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We show that a sectional-hyperbolic attracting set for a Hölder- $C^{1}$ vector field admits finitely many physical/SRB measures whose ergodic basins cover Lebesgue almost all points of the basin of topological attraction. In addition, these physical measures depend continuously on the flow in the $C^{1}$ topology, that is, sectional-hyperbolic attracting sets are statistically stable. To prove these results we show that each central-unstable disk in a neighborhood of this class of attracting sets is eventually expanded to contain a ball whose inner radius is uniformly bounded away from zero.

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“…The reason is that when Λ is a Lorenz attractor of vector fields in a Baire residual subset R r ⊂ X r (M 3 ), (r ∈ N ≥2 ) or Λ is a singular hyperbolic attractor Λ of vector fields in a Baire residual set R ⊂ X 1 (M ), then Λ is a homoclinic class such that each pair of periodic orbits are homoclinically related (cf. [5, Theorem 6.8] for Lorenz attractors and [20,Theorem B] for singular hyperbolic attractors) and…”

Section: (Lower Bound) Ifmentioning

confidence: 99%

On multifractal analysis and large deviations of singular-hyperbolic attractors

Shi¹,

Tian²,

Varandas³

et al. 2021

Preprint

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In this paper we study the multifractal analysis and large derivations for singular hyperbolic attractors, including the geometric Lorenz attractors. For each singular hyperbolic homoclinic class whose periodic orbits are all homoclinically related and such that the space of ergodic probability measures is connected, we prove that: (i) level sets associated to continuous observables are dense in the homoclinic class and satisfy a variational principle; (ii) irregular sets are either empty or are Baire generic and carry full topological entropy. The assumptions are satisfied by C 1generic singular hyperbolic attractors and C r -generic geometric Lorenz attractors (r ≥ 2). Finally we prove level-2 large deviations bounds for weak Gibbs measures, which provide a large deviations principle in the special case of Gibbs measures. The main technique we apply is the horseshoe approximation property.

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Robust transitivity of singular hyperbolic attractors

Cited by 13 publications

References 21 publications

Uniqueness of equilibrium states for Lorenz attractors in any dimension

Uniqueness of equilibrium states for Lorenz attractors in any dimension

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

On multifractal analysis and large deviations of singular-hyperbolic attractors

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