Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms

Kalinin, Boris; Sadovskaya, Victoria

doi:10.3934/dcds.2016.36.245

Cited by 12 publications

(8 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As the uniform invariant holonomies may not be unique (see, for example, [14]), we consider the cocycle and one of its holonomies in pairs. More precisely, H s α is the set of pairs ( Â, H s ) where Â ∈ S α ( ˆ , 2) and H s is a uniform stable holonomy for Â.…”

Section: Define the Setmentioning

confidence: 99%

Continuity of Lyapunov exponents for non-uniformly fiber-bunched cocycles

Freijo¹,

Marin²

2020

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

Section: Define the Setmentioning

confidence: 99%

Continuity of Lyapunov exponents for non-uniformly fiber-bunched cocycles

Freijo¹,

Marin²

2020

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

“…We note that Hölder continuity of the conjugacy with exponent less than β does not guarantee the intertwining, even for linear cocycles over hyperbolic systems; see [KaS16,Proposition 4.4] based on examples in [dlL92,NT98].…”

Section: Existence and Properties Of A Conjugacy Intertwining Holonomiesmentioning

confidence: 99%

“…Remark 3.2. Holonomies of a cocycle are sometimes defined as any family of homeomorphisms H A, s x,y satisfying properties (H1), (H2), (H4 0 ), and (H5) for nonlinear cocycles (see, for example, [ASV13]), and the holonomies H A,s as in Definition 1.3 are referred to as standard holonomies to distinguish them [KaS16]. Without condition (H3 0 ), uniqueness of holonomies may fail even for linear cocycles, as discussed in [KaS16] after Corollary 4.9.…”

Section: Existence and Properties Of The Holonomiesmentioning

confidence: 99%

Diffeomorphism cocycles over partially hyperbolic systems

Sadovskaya

2020

Ergod. Th. Dynam. Sys.

Self Cite

View full text Add to dashboard Cite

We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $\mathcal {M}$ . We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma }$ , we show that it has a continuous invariant family of $\gamma $ -Hölder Riemannian metrics on $\mathcal {M}$ . We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in $C^0$ for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle.

show abstract

“…Remark 2. If a linear cocycle is not fiber bunched, then it might admit multiple holonomies (see [KS1]).…”

Section: Definition 23 a Local Stable Holonomy For The Linear Cocyclesmentioning

confidence: 99%

Lyapunov spectrum properties and continuity of the lower joint spectral radius

Mohammadpour¹

2020

Preprint

View full text Add to dashboard Cite

In this paper, we study ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show the continuity of entropy spectrum at boundary of Lyapunov spectrum in the sense that h top (E(α t )) → h top (E(β(A)) for generic cocycles (in the sense of [BGMV]). We also show that for such cocycles over subshifts of finite type, the Lyapunov spectrum is equal to the closure of the set positive entropy spectrum. Moreover, we prove the restricted variational principle to level sets for such cocycles.We prove the continuity of the lower joint spectral radius for general cocycles under the assumption that the linear cocycles admit a dominated splitting of index 1.We show that the singular value pressure is continuous on Hölder and fiberbunched GL(2, R)-valued cocycles over subshifts of finite type. We also show that the Lyapunov spectrum is a closed and convex set for such cocycles over subshifts of finite type.

show abstract

Holonomies and cohomology for cocycles over partially hyperbolic diffeomorphisms

Cited by 12 publications

References 14 publications

Continuity of Lyapunov exponents for non-uniformly fiber-bunched cocycles

Continuity of Lyapunov exponents for non-uniformly fiber-bunched cocycles

Diffeomorphism cocycles over partially hyperbolic systems

Lyapunov spectrum properties and continuity of the lower joint spectral radius

Contact Info

Product

Resources

About