Gluing orbit property and partial hyperbolicity

Bomfim, Thiago; Torres, Maria Joana; Varandas, Paulo

doi:10.1016/j.jde.2020.09.040

Cited by 9 publications

(16 citation statements)

References 32 publications

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“…The latter relies ultimately on the uniqueness of equilibrium states for Hölder continuous potentials, a question which in such generality remains widely open. Moreover, singular hyperbolic attractors seem not to display any gluing orbit property, as hinted by [11,47,54]. We first show that the horseshoe approximation technique, valid for C r -generic geometric Lorenz attractors and C 1 -generic singular hyperbolic attractors, is enough to show that the level sets and irregular sets of such singular hyperbolic attractors inherit the properties of the corresponding objects for special classes of horseshoes approximating them.…”

Section: Introductionmentioning

confidence: 87%

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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Section: Introductionmentioning

confidence: 87%

On multifractal analysis and large deviations of singular-hyperbolic attractors

Shi¹,

Tian²,

Varandas³

et al. 2021

Preprint

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“…The gluing orbit property is a weakening of the notion of specification and briefly means that any finite pieces of orbits can be shadowed by a true orbit where the time lag between the pieces of orbits is bounded above by a constant that depends only on the shadowing distance. Such condition is embracing, as it is satisfied by transitive hyperbolic dynamics, minimal rotations on compact abelian groups and certain classes of partially hyperbolic diffeomorphisms [11,10,38]. Precise definitions will be given in the sequel.…”

Section: 34mentioning

confidence: 99%

Topological aspects of incompressible flows

Bessa

Torres

Varandas

2021

Journal of Differential Equations

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“…It is not hard to check that irrational rotations satisfy the gluing orbit property [9], but fail to satisfy the shadowing or specification properties. Partially hyperbolic examples exhibiting the same kind of behavior have been constructed in [10].…”

Section: And Letmentioning

confidence: 99%

On the rotation sets of generic homeomorphisms on the torus $\mathbb T^d$

Lima,

Varandas

2019

Preprint

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We study the rotation sets for homeomorphisms homotopic to the identity on the torus T d , d ≥ 2. In the conservative setting, we prove that there exists a Baire residual subset of the set Homeo 0,λ (T 2 ) of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in T 2 , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every d ≥ 2 the rotation set of C 0 -generic conservative homeomorphisms on T d is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.

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Gluing orbit property and partial hyperbolicity

Cited by 9 publications

References 32 publications

On multifractal analysis and large deviations of singular-hyperbolic attractors

On multifractal analysis and large deviations of singular-hyperbolic attractors

Topological aspects of incompressible flows

On the rotation sets of generic homeomorphisms on the torus $\mathbb T^d$

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