Definition:Hyperbolic Set of Riemannian Manifold

Definition

Let $M$ be a $C^1$ Riemannian manifold.

Let $U \subseteq M$ be a non-empty open subset.

Let $f : U \to f \sqbrk U$ be a $C^1$ diffeomorphism.

Let $\Lambda \subseteq U$ be a compact subset.

Suppose that $\Lambda$ is $f$-invariant such that:

$f \sqbrk \Lambda = \Lambda$

$\Lambda$ is hyperbolic if and only if there are:

$\lambda \in \openint 0 1$

$C > 0$

families $\map {E^s} x \subseteq T_x M$ and $\map {E^u} x \subseteq T_x M$

such that for each $x \in \Lambda$:

$(1):\quad T_x M = \map {E^s} x \oplus \map {E^u} x$

$(2):\quad \norm {\map {d f_x^n} {v^s} } \le C \lambda^n \norm {v^s}$ for every $v^s \in \map {E^s} x$ and $n \ge 0$

$(3):\quad \norm {\map {d f_x^{-n} } {v^u} } \le C \lambda^n \norm {v^u}$ for every $v^u \in \map {E^u} x$ and $n \ge 0$

$(4):\quad d f_x \sqbrk {\map {E^s} x} = \map {E^s} {\map f x}$ and $d f_x \sqbrk {\map {E^u} x} = \map {E^u} {\map f x}$

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$\map {E^s} x$ is called the stable subspace at $x$.

$\map {E^u} x$ is called the unstable subspace at $x$.

Sources

• 2002: Michael Brin and Garrett Stuck: Introduction to Dynamical Systems Chapter $5$: Hyperbolic Dynamics