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Ali Taghavi
Ali Taghavi
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Let $M$ be a manifold and $\phi_t$ be a smooth flow associated to a smooth vector field on $M$.

Are the following 3 conditions equivalent?

1)For every fixed $t$ the diffeomorphism $\phi_t$ is a hyperbolic diffeomorphism.(No continuity assumption on stable and unstable distributions as a grassman valued map in $t$)

2)The classical definition of hyperbolicityThe splitting of the tangent bundle to stable and unstable subspace(The splitting independent of t)

3)The condition $2$ with extra assumption that stable and unstable distributions are invariant under $D\phi_t$

Let $M$ be a manifold and $\phi_t$ be a smooth flow associated to a smooth vector field.

Are the following 3 conditions equivalent?

1)For every fixed $t$ the diffeomorphism $\phi_t$ is a hyperbolic diffeomorphism.(No continuity assumption on stable and unstable distributions as a grassman valued map in $t$)

2)The classical definition of hyperbolicityThe splitting of the tangent bundle to stable and unstable subspace(The splitting independent of t)

3)The condition $2$ with extra assumption that stable and unstable distributions are invariant under $D\phi_t$

Let $M$ be a manifold and $\phi_t$ be a smooth flow associated to a smooth vector field on $M$.

Are the following 3 conditions equivalent?

1)For every fixed $t$ the diffeomorphism $\phi_t$ is a hyperbolic diffeomorphism.(No continuity assumption on stable and unstable distributions as a grassman valued map in $t$)

2)The classical definition of hyperbolicityThe splitting of the tangent bundle to stable and unstable subspace(The splitting independent of t)

3)The condition $2$ with extra assumption that stable and unstable distributions are invariant under $D\phi_t$

Source Link
Ali Taghavi
Ali Taghavi
  • 356
  • 8
  • 31
  • 123

Some equivalent conditions for hyperbolicity of flow

Let $M$ be a manifold and $\phi_t$ be a smooth flow associated to a smooth vector field.

Are the following 3 conditions equivalent?

1)For every fixed $t$ the diffeomorphism $\phi_t$ is a hyperbolic diffeomorphism.(No continuity assumption on stable and unstable distributions as a grassman valued map in $t$)

2)The classical definition of hyperbolicityThe splitting of the tangent bundle to stable and unstable subspace(The splitting independent of t)

3)The condition $2$ with extra assumption that stable and unstable distributions are invariant under $D\phi_t$