Partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds and holonomy maps

Shi, Yi

doi:10.1016/j.crma.2014.07.002

Cited by 11 publications

(5 citation statements)

References 6 publications

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“…Let us observe that although for volume preserving partially hyperbolic diffeomorphism, accessibility implies transitivity, this is not true for dissipative partially hyperbolic diffeomorphisms, see the recent work of Y. Shi [46]. Y. Shi's work also shows that the assumption of non-vanishing exponent is necessary.…”

Section: Physical Measuresmentioning

confidence: 99%

Entropy along expanding foliations

Yang

2016

Preprint

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The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate of complexity generated by the diffeomorphism along the leaves of the foliation. We prove that this number varies upper semi-continuously with the diffeomorphism (C 1 topology), the invariant measure (weak* topology) and the foliation itself in a suitable sense.This has several important consequences. For one thing, it implies that the set of Gibbs u-states of C 1+ partially hyperbolic diffeomorphisms is an upper semi-continuous function of the map in the C 1 topology. Another consequence is that the sets of partially hyperbolic diffeomorphisms with mostly contracting or mostly expanding center are C 1 open. New examples of partially hyperbolic diffeomorphisms with mostly expanding center are provided, and the existence of physical measures for C 1 residual subset of diffeomorphisms are discussed.We also provide a new class of robustly transitive diffeomorphisms: every C 2 volume preserving, accessible partially hyperbolic diffeomorphism with one dimensional center and non-vanishing center exponent is C 1 robustly transitive (among neighborhood of diffeomorphisms which are not necessarily volume preserving).

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Section: Physical Measuresmentioning

confidence: 99%

Entropy along expanding foliations

Yang

2016

Preprint

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“…In particular we can also apply Theorem 1.2 and Theorem 4.3 to them. On the other hand, Shi [17] has announced that, on 3-nilmanifolds, there are partially hyperbolic diffeomorphism satisfying Axiom A. Of course, they have only one attractor and one repeller.…”

Section: Skew Productsmentioning

confidence: 99%

On the Three-Legged Accessibility Property

Hertz

Ures

2019

New Trends in One-Dimensional Dynamics

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We show that certain types of the three-legged accessibility property of a partially hyperbolic diffeomorphism imply the existence of a unique minimal set for one strong foliation and the transitivity of the other one. In case the center dimension is one, we also give a criteria to obtain three-legged accessibility in a robust way. We show some applications of our results to the time-one map of Anosov flows, skew products and certain Anosov diffeomorphisms with partially hyperbolic splitting.

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“…In those works it was established the relation between 11D supergravity reductions, by Sherk-Schwarz reductions [11][12][13][14], and geometric fluxes, also called torsion [15], or by more general fluxes described via embedding tensor mechanism [16,17]. String compactifications on twisted tori can be described in two complementary ways [18]: as a group manifold [19,20] (a nilmanifold [18,21,22]) or as T-duals of tori with constant NS-NS 3-form flux [23]. The…”

Section: Introductionmentioning

confidence: 99%

Fluxes, twisted tori, monodromy and U(1) supermembranes

Moral

Heras

León

et al. 2020

J. High Energ. Phys.

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We show that the D = 11 supermembrane theory (M2-brane) compactified on a M9× T2 target space, with constant fluxes C± naturally incorporates the geometrical structure of a twisted torus. We extend the M2-brane theory to a formulation on a twisted torus bundle. It is consistently fibered over the world volume of the M2-brane. It can also be interpreted as a torus bundle with a nontrivial U(1) connection associated to the fluxes. The structure group G is the area preserving diffeomorphisms. The torus bundle is defined in terms of the monodromy associated to the isotopy classes of symplectomorphisms with π0(G) = SL(2, Z), and classified by the coinvariants of the subgroups of SL(2, Z). The spectrum of the theory is purely discrete since the constant flux induces a central charge on the supersymmetric algebra and a modification on the Hamiltonian which renders the spectrum discrete with finite multiplicity. The theory is invariant under symplectomorphisms connected and non connected to the identity, a result relevant to guarantee the U-dual invariance of the theory. The Hamiltonian of the theory exhibits interesting new U(1) gauge and global symmetries on the worldvolume induced by the symplectomorphim transformations. We construct explicitly the supersymmetric algebra with nontrivial central charges. We show that the zero modes decouple from the nonzero ones. The nonzero mode algebra corresponds to a massive superalgebra that preserves either 1/2 or 1/4 of the original supersymmetry depending on the state considered.

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Partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds and holonomy maps

Cited by 11 publications

References 6 publications

Entropy along expanding foliations

Entropy along expanding foliations

On the Three-Legged Accessibility Property

Fluxes, twisted tori, monodromy and U(1) supermembranes

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