Boundedness, compactness, and invariant norms for banach cocycles over hyperbolic systems

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Abstract

We consider group-valued cocycles over dynamical systems with hyperbolic behavior. The base system is either a hyperbolic diffeomorphism or a mixing subshift of finite type. The cocycle A takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. We consider the periodic data of A, i.e., the set of its return values along the periodic orbits in the base. We show that if the periodic data of A is uniformly quasiconformal or bounded or contained in a compact set, then so is the cocycle. Moreover, in the latter case the cocycle is isometric with respect to a Hölder continuous family of norms. We also obtain a general result on existence of a measurable family of norms invariant under a cocycle.

Original languageEnglish (US)
Pages (from-to)3801-3812
Number of pages12
JournalProceedings of the American Mathematical Society
Volume146
Issue number9
DOIs
StatePublished - Jan 1 2018

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Banach spaces
Cocycle
Hyperbolic Systems
Stefan Banach
Compactness
Mathematical operators
Boundedness
Dynamical systems
Orbits
Norm
Invariant
Subshift
Quasiconformal
Diffeomorphism
Finite Type
Bounded Linear Operator
Compact Set
Isometric
Invertible
Periodic Orbits

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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abstract = "We consider group-valued cocycles over dynamical systems with hyperbolic behavior. The base system is either a hyperbolic diffeomorphism or a mixing subshift of finite type. The cocycle A takes values in the group of invertible bounded linear operators on a Banach space and is H{\"o}lder continuous. We consider the periodic data of A, i.e., the set of its return values along the periodic orbits in the base. We show that if the periodic data of A is uniformly quasiconformal or bounded or contained in a compact set, then so is the cocycle. Moreover, in the latter case the cocycle is isometric with respect to a H{\"o}lder continuous family of norms. We also obtain a general result on existence of a measurable family of norms invariant under a cocycle.",
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N2 - We consider group-valued cocycles over dynamical systems with hyperbolic behavior. The base system is either a hyperbolic diffeomorphism or a mixing subshift of finite type. The cocycle A takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. We consider the periodic data of A, i.e., the set of its return values along the periodic orbits in the base. We show that if the periodic data of A is uniformly quasiconformal or bounded or contained in a compact set, then so is the cocycle. Moreover, in the latter case the cocycle is isometric with respect to a Hölder continuous family of norms. We also obtain a general result on existence of a measurable family of norms invariant under a cocycle.

AB - We consider group-valued cocycles over dynamical systems with hyperbolic behavior. The base system is either a hyperbolic diffeomorphism or a mixing subshift of finite type. The cocycle A takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. We consider the periodic data of A, i.e., the set of its return values along the periodic orbits in the base. We show that if the periodic data of A is uniformly quasiconformal or bounded or contained in a compact set, then so is the cocycle. Moreover, in the latter case the cocycle is isometric with respect to a Hölder continuous family of norms. We also obtain a general result on existence of a measurable family of norms invariant under a cocycle.

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