Normally hyperbolic invariant manifold

Let M be a compact smooth manifold, f: M → M a diffeomorphism, and Df: TM → TM the differential of f. An f-invariant submanifold Λ of M is said to be a normally hyperbolic invariant manifold if the restriction to Λ of the tangent bundle of M admits a splitting into a sum of three Df-invariant subbundles, one being the tangent bundle of ${\displaystyle \Lambda }$, the others being the stable bundle and the unstable bundle and denoted E^s and E^u, respectively. With respect to some Riemannian metric on M, the restriction of Df to E^s must be a contraction and the restriction of Df to E^u must be an expansion, and must be relatively neutral on ${\displaystyle T\Lambda }$. Thus, there exist constants ${\displaystyle 0<\lambda <\mu ^{-1}<1}$ and c > 0 such that

${\displaystyle T_{\Lambda }M=T\Lambda \oplus E^{s}\oplus E^{u}}$

${\displaystyle (Df)_{x}E_{x}^{s}=E_{f(x)}^{s}{\text{ and }}(Df)_{x}E_{x}^{u}=E_{f(x)}^{u}{\text{ for all }}x\in \Lambda ,}$

${\displaystyle \|Df^{n}v\|\leq c\lambda ^{n}\|v\|{\text{ for all }}v\in E^{s}{\text{ and }}n>0,}$

${\displaystyle \|Df^{-n}v\|\leq c\lambda ^{n}\|v\|{\text{ for all }}v\in E^{u}{\text{ and }}n>0,}$

and

${\displaystyle \|Df^{n}v\|\leq c\mu ^{|n|}\|v\|{\text{ for all }}v\in T\Lambda {\text{ and }}n\in \mathbb {Z} .}$

• Stable manifold
• Center manifold
• Hyperbolic fixed point
• Hyperbolic set
• Hyperbolic Lagrangian coherent structures

• M.W. Hirsch, C.C Pugh, and M. Shub Invariant Manifolds, Springer-Verlag (1977), ISBN 978-3540081487 doi:10.1007/BFb0092042