# User:Neil Fenichel/Proposed/Normal hyperbolicity

## Inline Formulas

dynamical system on $$M_1$$

diffeomorphism $$F$$ of a manifold to itself or the flow $$F^t$$ defined by a vector field

point $$m^* = F^T(m)$$ that also is in $$M\ .$$ The derivative $$DF^T(m)$$

Dynamical system is $$C^r, r \geq 1\ ,$$ and

where $$\pi$$ is the projection onto $$N\ .$$ Let $$|| \ ||$$ be a metric on tangent vectors

Then the system of differential equations can be transformed to: $x' = ax$ $y' = by$ where $$a > 0$$ and $$b < 0\ .$$ I

The unstable manifold of $$P$$ is defined as the set of points $$(x, y)$$ near $$P$$ such that $$F^T(x, y, \varepsilon) \rightarrow P$$ as $$t \rightarrow -\infty\ .$$

the contraction factor of $$G^1$$ is approximately $$e^{(b-a)T}\ ,$$ and

The estimate of $$|\delta y|/|\delta x|$$ above

## Formulas for Exponential Rates

$v_{-t} = DF^{-t}(m) \cdot v_0$ and $$w_{-t} = \pi D F^{-t}(m) \cdot w_0$$

$\nu(m) = \inf \{a:(||w_0||/||w_{-t}||)/a^t \rightarrow 0$ as $$t \rightarrow \infty$$ for all $$w_0 \in N_m \}$$

$\sigma(m) = \inf \{ s: (||w_0||^s/||v_0||)/(||w_{-t}||^s/||v_{-t}||) \rightarrow 0$ as $$t \rightarrow \infty$$ for all $$v_0 \in T_m M$$ and $$w_0 \in N_m \} \ .$$

$\nu(m) < 1$ and $$\sigma(m) < 1/r$$ for all $$m \in M \ .$$

$\lambda^+(m) = \lim_{t \rightarrow \infty} ||\pi^+ DF^{-t}(m) | N^+ ||^{1/t}\ ,$ $\nu^-(m) = \overline{\lim_{t \rightarrow \infty}} ||\pi^- DF^t(F^{-t}(m)) | N^- ||^{1/t}\ ,$ $\sigma^-(m) = \overline{\lim_{t\rightarrow \infty}} \frac{\log || D(F^{-t} | M)(m)||}{-\log ||\pi^- DF^t(F^{-t}(m)) | N^- ||}\ .$

$\lambda^+(m) < 1$ and $$\nu^-(m) < 1$$ for all $$m \in M$$

$\sigma^-(m) < 1/r$ for all $$m \in M\ .$$